I have not found a proof of precisely this identity, but it follows quickly from the equations in <A HREF="http://www.ams.org/journals/proc/2001-129-03/S0002-9939-00-05821-4/S0002-9939-00-05821-4.pdf">
Orthogonal polynomials on the unit circle associated with the Laguerre polynomials.
</A> The Fourier cosine and sine transforms of $\phi_m$ are equal to the real and imaginary parts of $(1-z)z^m$, with $z=(2k-i)/(2k+i)$. According to equation (2.4) of that reference the desired integral is
$$\langle C\phi_m,S\phi_n\rangle=\frac{\pi}{2}\int_{|z|=1}[{\rm Re}\,(1-z)z^m][{\rm Im}\, (1-z)z^{n}]\frac{dz}{z}$$
*have to rush, more later this evening...*