We have $$J[\phi_1,\phi_2] = \int_{t_0}^{t_1} \left[ab^2 \dot{\phi}_1^2 + \frac{1}{2}ab^2 \dot{\phi}_2^2 + ab^2\phi_1\phi_2\cos(\phi_1 - \phi_2) \right] dt$$ where $\phi_1=\phi_1(t), \phi_2=\phi_2(t)$, $a,b$ constant.
The Euler-Lagrange equations for several functions of a single variable are $\displaystyle 0= \frac{\partial f}{\partial \phi_i} - \frac{\partial}{\partial t}\frac{\partial f}{\partial\dot{\phi}_i},$ where $$f(t,\phi_1, \phi_2, \dot{\phi}_1, \dot{\phi}_2) =ab^2 \dot{\phi}_1^2 + \frac{1}{2}ab^2 \dot{\phi}_2^2 + ab^2\phi_1\phi_2\cos(\phi_1 - \phi_2).$$
I have \begin{align} 0 &= \frac{\partial f}{\partial \phi_1} - \frac{\partial}{\partial t}\frac{\partial f}{\partial \dot{\phi}_1}\\ &= ab^2\phi_2\cos(\phi_1 - \phi_2) - ab^2\phi_1\phi_2\sin(\phi_1 - \phi_2) - \frac{\partial}{\partial t}\left( 2ab^2 \dot{\phi}_1 \right)\\ &= ab^2\phi_2\cos(\phi_1 - \phi_2) - ab^2 \phi_1\phi_2\sin(\phi_1 - \phi_2) - 2ab^2\ddot{\phi}_1\\ 0 &= \frac{\partial f}{\partial \phi_2} - \frac{\partial}{\partial t}\frac{\partial f}{\partial\dot{\phi}_2}\\ &= ab^2\phi_1\cos(\phi_1-\phi_2)+ab^2\phi_1\phi_2\sin(\phi_1 - \phi_2)-\frac{\partial}{\partial t}(ab^2 \dot{\phi}_2)\\ &= ab^2\phi_1\cos(\phi_1 - \phi_2)+ab^2\phi_1\phi_2\sin(\phi_1 - \phi_2)-ab^2 \ddot{\phi}_2 \end{align} as the system of 2 second order ode's: \begin{align*} 0&=\phi_2\cos(\phi_1 - \phi_2) - \phi_1\phi_2\sin(\phi_1 - \phi_2) - 2\ddot{\phi}_1 \\ 0&=\phi_1\cos(\phi_1 - \phi_2)+\phi_1\phi_2\sin(\phi_1 - \phi_2)- \ddot{\phi}_2. \end{align*}
Am I on the right track? Are there two EL equations, one for each $\phi_j?$
Yes, you are on the right track, and there are two EL equations, one for each $\phi_j.$ I don't envy you solving that system of ODE's, as it's highly non-linear and coupled.