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I don't have know a reference off hand. So let me just give you a proof when $char\, k=0$. I expect it's true in general.

We can assume without loss of generality that $k=\mathbb{C}$. Then via the exponential sequence and the Lefschetz $(1,1)$ theorem, $NS(X)_\mathbb{Q}$ (your $NS(X)$) can be identified the space of $(1,1)$ classes in $H^2(X,\mathbb{Q})$. Given a nonzero $D\in NS(X)_\mathbb{Q}$, there exists $D''\in H^{d-1,d-1}(X)\cap H^{2d-2}(X,\mathbb{Q})$ such that $D\cup D''\not=0$ by Poincare duality. Hard Lefschetz tells us that $D'' = \Theta^{d-2}\cdot D'$ for some $D'\in NS(X)_\mathbb{Q}$.

Here's a different argument which works over any algebraically closed field. Let $Y\subset X$ be surface given as intersection of $d-2$ general hyperplanes with respect to the embedding given by $N\Theta$, $N\gg 0$.

Claim: The restriction $NS(X)_\mathbb{Q}\to NS(Y)_\mathbb{Q}$ is injective.

Proof: We can see, from the Kummer sequence, that $NS(X)_\mathbb{Q}\subset NS(X)_\mathbb{Q_\ell}$ embeds into $\ell$-adic cohomology $H^2(X_{et},\mathbb{Q}_\ell)$. Therefore the claim follows from weak Lefschetz [Milne, Etale cohomology]

The pairing on $NS(X)_\mathbb{Q}$ is nonzero multiple of the restriction of the pairing on $Y$. The result now follows from the Hodge index theorem for surfaces [Hartshorne, Alg. Geom.]

$\DeclareMathOperator\NS{NS}$I don't have know a reference off hand. So let me just give you a proof when $\operatorname{char} k=0$. I expect it's true in general.

We can assume without loss of generality that $k=\mathbb{C}$. Then via the exponential sequence and the Lefschetz $(1,1)$ theorem, $\NS(X)_\mathbb{Q}$ (your $\NS(X)$) can be identified the space of $(1,1)$ classes in $H^2(X,\mathbb{Q})$. Given a nonzero $D\in \NS(X)_\mathbb{Q}$, there exists $D''\in H^{d-1,d-1}(X)\cap H^{2d-2}(X,\mathbb{Q})$ such that $D\cup D''\ne0$ by Poincare duality. Hard Lefschetz tells us that $D'' = \Theta^{d-2}\cdot D'$ for some $D'\in \NS(X)_\mathbb{Q}$.

Here's a different argument which works over any algebraically closed field. Let $Y\subset X$ be a surface given as the intersection of $d-2$ general hyperplanes with respect to the embedding given by $N\Theta$, $N\gg 0$.

Claim: The restriction $\NS(X)_\mathbb{Q}\to \NS(Y)_\mathbb{Q}$ is injective.

Proof: We can see, from the Kummer sequence, that $\NS(X)_\mathbb{Q}\subset \NS(X)_\mathbb{Q_\ell}$ embeds into $\ell$-adic cohomology $H^2(X_{\text{ét}},\mathbb{Q}_\ell)$. Therefore the claim follows from weak Lefschetz [Milne, Etale cohomology].

The pairing on $\NS(X)_\mathbb{Q}$ is a nonzero multiple of the restriction of the pairing on $Y$. The result now follows from the Hodge index theorem for surfaces [Hartshorne, Alg. Geom.].

I don't have know a reference off hand. So let me just give you a proof when $char\\, k=0$. I expect it's true in general.

We can assume without loss of generality that $k=\mathbb{C}$. Then via the exponential sequence and the Lefschetz $(1,1)$ theorem, $NS(X)_\mathbb{Q}$ (your $NS(X)$) can be identified the space of $(1,1)$ classes in $H^2(X,\mathbb{Q})$. Given a nonzero $D\in NS(X)_\mathbb{Q}$, there exists $D''\in H^{d-1,d-1}(X)\cap H^{2d-2}(X,\mathbb{Q})$ such that $D\cup D''\not=0$ by Poincare duality. Hard Lefschetz tells us that $D'' = \Theta^{d-2}\cdot D'$ for some $D'\in NS(X)_\mathbb{Q}$.

Here's a different argument which works over any algebraically closed field. Let $Y\subset X$ be surface given as intersection of $d-2$ general hyperplanes with respect to the embedding given by $N\Theta$, $N\gg 0$.

Claim: The restriction $NS(X)_\mathbb{Q}\to NS(Y)_\mathbb{Q}$ is injective.

Proof: We can see, from the Kummer sequence, that $NS(X)_\mathbb{Q}\subset NS(X)_\mathbb{Q_\ell}$ embeds into $\ell$-adic cohomology $H^2(X_{et},\mathbb{Q}_\ell)$. Therefore the claim follows from weak Lefschetz [Milne, Etale cohomology]

The pairing on $NS(X)_\mathbb{Q}$ is nonzero multiple of the restriction of the pairing on $Y$. The result now follows from the Hodge index theorem for surfaces [Hartshorne, Alg. Geom.]

I don't have know a reference off hand. So let me just give you a proof when $char\, k=0$. I expect it's true in general.

We can assume without loss of generality that $k=\mathbb{C}$. Then via the exponential sequence and the Lefschetz $(1,1)$ theorem, $NS(X)_\mathbb{Q}$ (your $NS(X)$) can be identified the space of $(1,1)$ classes in $H^2(X,\mathbb{Q})$. Given a nonzero $D\in NS(X)_\mathbb{Q}$, there exists $D''\in H^{d-1,d-1}(X)\cap H^{2d-2}(X,\mathbb{Q})$ such that $D\cup D''\not=0$ by Poincare duality. Hard Lefschetz tells us that $D'' = \Theta^{d-2}\cdot D'$ for some $D'\in NS(X)_\mathbb{Q}$.

Here's a different argument which works over any algebraically closed field. Let $Y\subset X$ be surface given as intersection of $d-2$ general hyperplanes with respect to the embedding given by $N\Theta$, $N\gg 0$.

Claim: The restriction $NS(X)_\mathbb{Q}\to NS(Y)_\mathbb{Q}$ is injective.

Proof: We can see, from the Kummer sequence, that $NS(X)_\mathbb{Q}\subset NS(X)_\mathbb{Q_\ell}$ embeds into $\ell$-adic cohomology $H^2(X_{et},\mathbb{Q}_\ell)$. Therefore the claim follows from weak Lefschetz [Milne, Etale cohomology]

The pairing on $NS(X)_\mathbb{Q}$ is nonzero multiple of the restriction of the pairing on $Y$. The result now follows from the Hodge index theorem for surfaces [Hartshorne, Alg. Geom.]

I don't have a good reference off hand. So let me just give you a proof when $char\\, k=0$. I expect it's true in general.

We can assume without loss of generality that $k=\mathbb{C}$. Then via the exponential sequence and the Lefschetz $(1,1)$ theorem, $NS(X)_\mathbb{Q}$ (your $NS(X)$) can be identified the space of $(1,1)$ classes in $H^2(X,\mathbb{Q})$. Given a nonzero $D\in NS(X)_\mathbb{Q}$, there exists $D''\in H^{d-1,d-1}(X)\cap H^{2d-2}(X,\mathbb{Q})$ such that $D\cup D''\not=0$ by Poincare duality. Hard Lefschetz tells us that $D'' = \Theta^{d-2}\cdot D'$ for some $D'\in NS(X)_\mathbb{Q}$.

I don't have know a reference off hand. So let me just give you a proof when $char\\, k=0$. I expect it's true in general.

We can assume without loss of generality that $k=\mathbb{C}$. Then via the exponential sequence and the Lefschetz $(1,1)$ theorem, $NS(X)_\mathbb{Q}$ (your $NS(X)$) can be identified the space of $(1,1)$ classes in $H^2(X,\mathbb{Q})$. Given a nonzero $D\in NS(X)_\mathbb{Q}$, there exists $D''\in H^{d-1,d-1}(X)\cap H^{2d-2}(X,\mathbb{Q})$ such that $D\cup D''\not=0$ by Poincare duality. Hard Lefschetz tells us that $D'' = \Theta^{d-2}\cdot D'$ for some $D'\in NS(X)_\mathbb{Q}$.

Here's a different argument which works over any algebraically closed field. Let $Y\subset X$ be surface given as intersection of $d-2$ general hyperplanes with respect to the embedding given by $N\Theta$, $N\gg 0$.

Claim: The restriction $NS(X)_\mathbb{Q}\to NS(Y)_\mathbb{Q}$ is injective.

Proof: We can see, from the Kummer sequence, that $NS(X)_\mathbb{Q}\subset NS(X)_\mathbb{Q_\ell}$ embeds into $\ell$-adic cohomology $H^2(X_{et},\mathbb{Q}_\ell)$. Therefore the claim follows from weak Lefschetz [Milne, Etale cohomology]

The pairing on $NS(X)_\mathbb{Q}$ is nonzero multiple of the restriction of the pairing on $Y$. The result now follows from the Hodge index theorem for surfaces [Hartshorne, Alg. Geom.]