A TRACON Controller reported they issued a vector to an aircraft which placed it below the Minimum Vectoring Altitude.

Narrative

Aircraft X called up off ZZZ level at 3600 feet for an IFR clearance to GRF via V204; the MVA in that area was 3700 feet. I asked them if they could maintain VFR through 5000 feet for their IFR clearance; they indicated they could through 4000 feet. I told them leaving 4000 ft. cleared to GRF via radar vectors and maintain 10000 ft.; their filed altitude. The vector was to join the airway but I confused V4 for V204. The aircraft appeared to level at 4000 ft. which I instructed them to maintain 10000 ft. They complied. They went through V4; which I was thinking was V204. At that point they were at about 5000 ft. in a 5500 ft. MVA heading toward a 6300 ft. MVA. I told them they were in an area of higher terrain and the minimum vectoring altitude was 6300 ft. and asked if they had the terrain in sight; they replied that they did. Aircraft X continued to climbing above the MVA altitude and joined V204. In this event I should have had a better knowledge of the airways. I confused V204 with V4. I anticipated the aircraft's climb more than I should have and also should have vectored the aircraft south to stay in lower terrain until getting higher to join the airway to the southwest.