You need to do this via a local argument. A good reference for local components of $\mathrm{GL}_2 \times \mathrm{GL}_2$ automorphic representations is Gelbart and Jacquet, "[A relation between automorphic representations of $\mathrm{GL}(2)$ and $\mathrm{GL}(3)$][1]". For just the $\mathrm{GL}_2$ theory, see Schmidt, "[Some remarks on local newforms for $\mathrm{GL}(2)$][2]".

 - At all primes $v \nmid pq$, the local epsilon factors and conductor exponents are trivial.
 - The local component of $f$ at $p$ is a principal series representation $\pi_{f,p} = \omega_{f,p,1} \boxplus \omega_{f,p,2}$, where $\omega_{f,p,1}, \omega_{f,p,2}$ are character of $\mathbb{Q}_p^{\times}$ of conductor exponent $c(\omega_{f,p,1}) = 1$ and $c(\omega_{f,p,2}) = 0$, while the local component of $g$ is a spherical principal series representation $\pi_{g,p} = \omega_{g,p,1} \boxplus \omega_{g,p,2}$ with both characters unramified, so that $c(\omega_{g,p,1}) = c(\omega_{g,p,2}) = 0$. Then
\\[\pi_{f,p} \otimes \pi_{g,p} = \omega_{f,p,1} \omega_{g,p,1} \boxplus \omega_{f,p,1} \omega_{g,p,2} \boxplus \omega_{f,p,2} \omega_{g,p,1} \boxplus \omega_{f,p,2} \omega_{g,p,2}.\\]
The conductor exponent is
\\[c(\pi_{f,p} \otimes \pi_{g,p}) = c(\omega_{f,p,1} \omega_{g,p,1}) + c(\omega_{f,p,1} \omega_{g,p,2}) + c(\omega_{f,p,2} \omega_{g,p,1}) + c(\omega_{f,p,2} \omega_{g,p,2}),\\]
which is
\\[1 + 1 + 0 + 0 = 2.\\]
The epsilon factor $\epsilon_p(s,\pi_{f,p} \otimes \pi_{g,p},\psi_p)$ ostensibly depends on an additive character $\psi_p$ of $\mathbb{Q}_p$, which we may choose to be unramified (i.e. $c(\psi_p) = 0$), though the global epsilon factor is independent of this. Anyway, $\epsilon_p(s,\pi_{f,p} \otimes \pi_{g,p},\psi_p)$ is equal to
\\[\epsilon_p(s,\omega_{f,p,1} \omega_{g,p,1},\psi_p) \epsilon_p(s,\omega_{f,p,1} \omega_{g,p,2},\psi_p) \epsilon_p(s,\omega_{f,p,2} \omega_{g,p,1},\psi_p) \epsilon_p(s,\omega_{f,p,2} \omega_{g,p,2},\psi_p),\\]
which is
\\[\left(\omega_{g,p,1}(p) \epsilon_p(s,\omega_{f,p,1},\psi_p)\right) \cdot \left(\omega_{g,p,2}(p) \epsilon_p(s,\omega_{f,p,1},\psi_p)\right) \cdot 1 \cdot 1 = \epsilon_p(s,\pi_{f,p},\psi_p)^2.\\]
(See Proposition 1.4 of Gelbart and Jacquet and equations (4) and (6) of Schmidt and use the fact that $g$ has principal nebentypus means that the central character $\omega_{\pi_{g,p}} = \omega_{g,p,1} \omega_{g,p,2}$ of $\pi_{g,p}$ is trivial, so that $\omega_{g,p,1}(p) \omega_{g,p,2}(p) = 1$.) As the conductor exponent is $2$,
\\[\epsilon_p(s,\pi_{f,p} \otimes \pi_{g,p},\psi_p) = \epsilon_p\left(\frac{1}{2},\pi_{f,p} \otimes \pi_{g,p},\psi_p\right) p^{-2\left(\frac{1}{2} - s\right)}.\\]
 - The local components of $f$ and $g$ at $q$ are special representations $\pi_{f,q} = \omega_{f,q} \mathrm{St}_q$  $\pi_{g,q} = \omega_{g,q} \mathrm{St}_q$ with $\omega_{f,q}, \omega_{g,q}$ characters of $\mathbb{Q}_q^{\times}$ that are unramified and either trivial or quadratic. The conductor exponent of $c(\pi_{f,q} \otimes \pi_{g,q})$ is $2$, as $\pi_{f,q} \otimes \pi_{g,q}$ is the isobaric sum of an unramified character of $\mathrm{GL}_1(\mathbb{Q}_p)$ and the Steinberg representation of $\mathrm{GL}_3(\mathbb{Q}_p)$, with the conductor exponent of the former being $0$ and the latter being $2$. By Proposition 1.4 of Gelbart and Jacquet and equation (11) of Schmidt, the epsilon factor $\epsilon_q(s,\pi_{f,q} \otimes \pi_{g,q},\psi_q)$ is
\\[\epsilon_q\left(s + \frac{1}{2},\omega_{g,q} \omega_{f,q} \mathrm{St}_q, \psi_q\right) \epsilon_q\left(s - \frac{1}{2},\omega_{g,q} \omega_{f,q} \mathrm{St}_q, \psi_q\right) = \epsilon_q(s,\omega_{g,q} \omega_{f,q} \mathrm{St}_q, \psi_q)^2.\\]
One can further show that this is equal to
\\[\epsilon_q(s,\pi_{g,q}, \psi_q)^2 = \epsilon_q(s,\pi_{f,q}, \psi_q)^2 = p^{-2(\frac{1}{2} - s)}\\]
(the reference for this is Section 11.12 of Goldfeld and Hundley's book combined with Schmidt's paper).
 - Finally, the local component of $f$ at $\infty$ is the discrete series representation $\pi_{f,\infty} = D_{k-1}$ of weight $k$, while $\pi_{g,\infty} = D_{k_g - 1}$. Then $\pi_{f,\infty} \otimes \pi_{g,\infty} = D_{|k - k_g|} \boxplus D_{k + k_g}$, and so
\\[\epsilon_{\infty}(s, \pi_{f,\infty} \otimes \pi_{g,\infty},\psi_{\infty}) = \epsilon_{\infty}(s,D_{|k - k_g|},\psi_{\infty}) \epsilon_{\infty}(s,D_{k + k_g},\psi_{\infty}),\\]
which is
\\[i^{|k - k_g| + 1} i^{k + k_g + 1} = (-1)^{\max\\{k,k_g\\}+1}.\\]
The best reference for this is Knapp, "[Local Langlands Correspondence: The Archimedean Case][3]".

So defining the global epsilon factor
\\[\epsilon(s,\pi_f \otimes \pi_g) = \epsilon_{\infty}(s, \pi_{f,\infty} \otimes \pi_{g,\infty},\psi_{\infty}) \prod_{p'} \epsilon_{p'}(s, \pi_{f,p'} \otimes \pi_{g,p'},\psi_{p'}),\\]
where $p'$ runs over all primes, the functional equation for the completed Rankin-Selberg $L$-function $\Lambda(s, \pi_f \otimes \pi_g)$ is
\\[\Lambda(s, \pi_f \otimes \pi_g) = \epsilon(s,\pi_f \otimes \pi_g) \Lambda(1 - s, \widetilde{\pi}_f \otimes \widetilde{\pi}_g).\\]
By the above discussion,
\\[\epsilon(s,\pi_f \otimes \pi_g) = (pq)^2 (-1)^{\max\\{k,k_g\\}+1} \epsilon_p\left(\frac{1}{2},\pi_{f,p},\psi_p\right)^2.\\]
Finally, we note that $\epsilon_p\left(\frac{1}{2},\pi_{f,p},\psi_p\right) = \eta_f(p)$, the pseudo-eigenvalue of $f$ corresponding to the Atkin-Lehner operator $W_p$; see equation (7.6) of Templier, "[Voronoï Summation for $\mathrm{GL}(2)$][4]", and equation (7.8) shows that
\\[\eta_f(p) = \overline{\lambda_f}(p) \tau(\chi_p) p^{-1/2}\\]
(note that $|\lambda_f(p)| = 1$).


  [1]: https://dx.doi.org/10.24033/asens.1355
  [2]: http://www2.math.ou.edu/~rschmidt/papers/gl2.pdf
  [3]: https://www.math.stonybrook.edu/~aknapp/pdf-files/motives.pdf
  [4]: http://www.math.cornell.edu/~templier/papers/voronoi-gl2.pdf