Assume there is a solution where $ Y, K, C$ all have constant growth rates.
Impose that solution on Euler and you will have $Y/K$ is constant i.e growth rates of $Y$ and $K$ are same.
Use the production function and you will have growth rate of $Y = \gamma_x + n $ where $n$ is the growth rate of $N$.
Then use the budget constraint (replacing $I$) and you will find that $C/K$ is not a constant because of $q$.
Thus if a solution exists where output, capital and consumption all have constant growth rates, in such a solution output and capital grow at the same rate which is different from that of consumption.
I deliberately did not show how to manipulate the production function and budget constraint - the first time I worked on this type of problem, figuring out the manipulation was fun; perhaps you will also enjoy it :)
Don’t take logs and differentiate - your notation suggests you have discrete time. :)