(a) Prove that there exist infinitely many values of integer parameter $a$ such that equation $$ 2 x^2+a x y+y^2+1 = 0 $$ is solvable in integers $(x,y)$.

(b) The same question for a similar equation $$ 2 x^2+a x y -y^2-1 = 0. $$

Here is a conditional proof. We will look for solutions with $x$ even. Considering the equations as quadratic in $y$, we need the determinant $(ax)^2\pm 8x^2-4$ to be a perfect square of an even integer, say $(2t)^2$. Equivalently, $$ t^2-(a^2\pm 8)(x/2)^2=-1. $$
Hence, we need to prove that there are infinitely many integers $d$ of the form $d=a^2\pm 8$ such that the negative Pell equation $t^2-du^2=-1$ is solvable. It is known to be solvable if, for example, $d$ is a prime of the form $4k+1$. By Bunyakovsky conjecture, $a^2+8$ (and similarly $a^2-8$) is a prime infinitely often. Obviously, if this is a prime then $a$ is odd, hence prime $a^2\pm 8$ is of the form $4k+1$, and the result follows.

I suspect that there should be a not-too-difficult unconditional proof, but currently do not see it, hence the question.

(a) Prove that there exist infinitely many values of integer parameter $a$ such that equation $$ 2 x^2+a x y+y^2+1 = 0 $$ is solvable in integers $(x,y)$.

(b) The same question for a similar equation $$ 2 x^2+a x y -y^2-1 = 0. $$

Here is a conditional proof. We will look for solutions with $x$ even. Considering the equations as quadratic in $y$, we need the determinant $(ax)^2\pm 8x^2-4$ to be a perfect square of an even integer, say $(2t)^2$. Equivalently, $$ t^2-(a^2\pm 8)(x/2)^2=-1. $$
Hence, we need to prove that there are infinitely many integers $d$ of the form $d=a^2\pm 8$ such that the negative Pell equation $t^2-du^2=-1$ is solvable. It is known to be solvable if, for example, $d$ is a prime of the form $4k+1$. By Bunyakovsky conjecture, $a^2+8$ (and similarly $a^2-8$) is a prime infinitely often. Obviously, if this is a prime then $a$ is odd, hence prime $a^2\pm 8$ is of the form $4k+1$, and the result follows.

I suspect that there should be a not-too-difficult unconditional proof, but currently do not see it, hence the question.

An update inspired by Stanley Yao Xiao's answer. In fact, there is a theorem proved in

Newman, M. (1977) A note on an equation related to the Pell equation, Amer. Math. Monthly, 84, 365–366.

stating that if $d = \prod_{i=1}^r p_i$, where $r=2$ or $r$ is odd, and all $p_i$ are primes congruent to $1$ modulo $4$ satisfying $\left(\frac{p_i}{p_j}\right)=-1$ for $i\neq j$, then the negative Pell equation $t^2-du^2=-1$ is solvable in integers. Hence, it would suffices to show that $a^2 \pm 8$ is of this form infinitely often. Stanley Yao Xiao's answer corresponds to the case $r=2$ of this observation. As correctly noted in the answer, the difficulty of this approach is in the condition $\left(\frac{p_i}{p_j}\right)=-1$.