Connected Vertex Cover
(First, a side note: it happens that Connected Vertex Cover isn't actually NP-hard for all classes of graphs. In particular, it's solvable in polynomial time in chordal graphs and graphs with maximum degree 3. However, it is NP-hard for planar graphs with maximum degree 4 in particular, and that's the easiest case to prove.)

For this, you'll want to reduce from Planar Vertex Cover. This is the special case of Vertex Cover where the input graph is planar, and it itself can be proved NP-hard by reduction from Planar 3Sat. (Planar 3Sat is, similarly, the special case of 3Sat where the input graph is planar. Proved NP-hard by reduction from normal 3Sat, and it itself is used to prove that various games are NP-hard, such as Super Mario Bros, Super Mario Bros 3, and Super Mario World.)

This reduction was first done by Garey and Johnson in 1977. I'm currently trying to find the full details of their proof, but these lecture notes provide a general overview.