B737 pilot reported while in cruise flight experiencing engine oil quantity decreasing resulting in the flight diverting to an alternate airport.

Narrative

After initial level off at FL350; FO unblanked the engine instruments and noted that #1 oil quantity had decreased 2qt (from 17 to 15) since the departure gate. 15 minutes later; quantity dropped to 14. I sent an ACARS and called dispatch/maintenance control to discuss min acceptable for us to continue over water to ZZZZ. During the conversation; #1 oil quantity dropped to 13qt while #2 oil remained stable at 21qt. We decided that running out of oil over the ocean was a non-optimal situation; so we agreed on a divert to ZZZ. ZZZ1 was also an option; but we didn't think we'd retain enough oil to make it there at the 1qt per 15-20 min rate that it was decreasing. Aircraft weight at this point was about 163;000# (10;000# above max structural landing weight).We prepared for an overweight landing (FOM chXX) and made a soft landing on the longest runway (XX). We rolled to the end using minimal braking and had the Airport Rescue and Firefighting (ARFF) check brake temperatures before taxiing to the gate. ARFF followed us to the gate and took another brake temperature reading when we parked. ARFF confirmed that brake temperatures were decreasing and not a problem. The aircraft deplaned normally.Cause: unknown oil leakOnly recommendation is to closely monitor oil consumption and keep high consumers away from long over water flights.