What is the complexity of k-clique problem with a predetermined vertex in the solution?
It is still $\mathsf{NP}$-complete.
Consider the following reduction from the normal $\mathrm{Clique}$ problem:
Given a graph $G$ and some desired clique size $k$, add a new vertex $v^\ast$ which is connected to all vertices of $G$ to obtain a new graph $G^\ast$.
Then $(G^\ast, k + 1, v^\ast)$ is a yes-instance of your modified $\mathrm{Clique}$ problem if and only if $(G, k)$ is a yes-instance of $\mathrm{Clique}$.