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 $\mathrm{GL}(3)$". For just the $\mathrm{GL}_2$ theory, see Schmidt, "Some remarks on local newforms for $\mathrm{GL}(2)$".

- 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".

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 $\mathrm{GL}(2)$", and equation (7.8) shows that
\[\eta_f(p) = \overline{\lambda_f}(p) \tau(\chi_p) p^{-1/2}\]
(note that $|\lambda_f(p)| = 1$).