modify Dijkstra's algorithm to compute shortest path only for the vertex which is no more than three edges away from the start vertex
I assume you want to only calculate shortest paths, starting from $s$, to vertices in its 3-hop neighborhood, i.e. vertices which are reachable from the start vertex $s$ with a path length $\leq 3$.

Use a limited breadth-first search and save the frontier vertices in a queue which is sorted according to its edge-weights. By doing that you have a priority queue and can update vertices like in the regular Dijkstra's algorithm. Note that you need to keep track of the current depth of the breadth-first search.