The answer is no in general: for example, if $\phi$ is a cuspidal automorphic form in a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$ and $\Phi$ is an Eisenstein series in the noncuspidal automorphic representation $\Pi = \widetilde{\pi} \boxplus \omega$ of $\mathrm{GL}_n(\mathbb{A}_F)$, where $\omega$ is a Hecke character, then were this integral to converge absolutely, it would represent $L(s,\Pi \times \pi) = L(s,\widetilde{\pi} \times \pi) L(s,\pi \otimes \omega)$. However, this has a pole at $s = 1$.

On the other hand, if $\Phi$ is cuspidal, then this integral converges absolutely even if $\phi$ is an Eisenstein series. More generally, in order to relate this integral to $L$-functions, you need to use a regularisation process, which is due to Ichino and Yamana:

https://doi.org/10.1112/S0010437X14007362

The idea is to use a truncation operator on these automorphic forms that truncates up to height $T$, and then view this as a polynomial in $T$. The constant term in this polynomial is precisely what one would hope for, namely $$\int\limits_{\mathrm{N}_n(\mathbb{A}_F) \backslash \mathrm{GL}_n(\mathbb{A}_F)} W_{\Phi}\begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} W_{\phi}(g) \left|\det g\right|^{s - \frac{1}{2}} \, dg,$$ where $W_{\Phi} \in \mathcal{W}(\Pi,\psi)$ is the Whittaker function associated to $\Phi$ and $W_{\phi} \in \mathcal{W}(\pi,\overline{\psi})$ is the Whittaker function associated to $\phi$. If these Whittaker functions are pure tensors, then this global integral factorises as a product of local integrals, each of which represents the local component of the $\mathrm{GL}_{n + 1} \times \mathrm{GL}_n$ Rankin-Selberg $L$-function.

The answer is no in general: for example, if $\phi$ is a cuspidal automorphic form in a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$ and $\Phi$ is an Eisenstein series in the noncuspidal automorphic representation $\Pi = \widetilde{\pi} \boxplus \omega$ of $\mathrm{GL}_n(\mathbb{A}_F)$, where $\omega$ is a Hecke character, then were the integral $I(s,\Phi,\phi)$ to converge absolutely, it would represent $L(s,\Pi \times \pi) = L(s,\widetilde{\pi} \times \pi) L(s,\pi \otimes \omega)$. However, this has a pole at $s = 1$.

On the other hand, if $\Phi$ is cuspidal, then the integral $I(s,\Phi,\phi)$ converges absolutely even if $\phi$ is an Eisenstein series. More generally, in order to relate $I(s,\Phi,\phi)$ to $L$-functions, you need to use a regularisation process, which is due to Ichino and Yamana:

https://doi.org/10.1112/S0010437X14007362

The idea is to use a truncation operator on these automorphic forms that truncates up to height $T$, and then view the integral involving these truncated automorphic forms as a polynomial in $T$. The constant term in this polynomial is precisely what one would hope for, namely $$\int\limits_{\mathrm{N}_n(\mathbb{A}_F) \backslash \mathrm{GL}_n(\mathbb{A}_F)} W_{\Phi}\begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} W_{\phi}(g) \left|\det g\right|^{s - \frac{1}{2}} \, dg,$$ where $W_{\Phi} \in \mathcal{W}(\Pi,\psi)$ is the Whittaker function associated to $\Phi$ and $W_{\phi} \in \mathcal{W}(\pi,\overline{\psi})$ is the Whittaker function associated to $\phi$.

Indeed, if $\Phi$ is cuspidal, then $I(s,\Phi,\phi)$ is equal to this integral involving Whittaker functions, which is proven by inserting the Whittaker expansion of $\Phi$ and unfolding. When $\Phi$ is not cuspidal, however, this unfolding process will lead to the presence of certain degenerate integrals, and these integrals need not converge (and in fact do not converge due to exponential growth, as discussed in the introduction of Ichino and Yamana's paper).

Finally, if the Whittaker functions $W_{\Phi}$ and $W_{\phi}$ are both pure tensors, then this global integral involving these Whittaker functions factorises as a product of local integrals, each of which represents the local component of the completed $\mathrm{GL}_{n + 1} \times \mathrm{GL}_n$ Rankin-Selberg $L$-function $\Lambda(s,\Pi \times \pi)$.