B737-900 Captain reported fuel quantity systems indications were unreliable and erratic during ETOPS operations. The Flight Crew ran the QRH and checklists; but were unable to solve the problem. The Flight Crew elected to continue flight to destination airport.

Narrative

We departed the Gate with main tanks full and remaining fuel in the Center Tank for a 24;100 pounds total fuel on board. During climb out from ZZZ; approximately 3 minutes after departure we discovered a fuel imbalance of approximately 680 pounds. The left main tank was low approximately at 7;920 pounds. The Center Tank still had over 4;000 pounds and the right main tank indicating approximately 8;610 pounds. At level off we watched the fuel and the left main tank seemed to stabilize at 7;920 pounds while we burned the center tank empty. After the center tank is empty we then Cross Feed to balance the main tanks. Monitoring our fuel burn with the HOWGOZIT we started to show an over burn. At one point comparing our FMC Progress page fuel report for positions passed we had over burned by 1;200 pounds. This had me very concerned sense we are now about in the middle of the ZZZ1 between ZZZ and ZZZ2. I send a ELB write up and on the single page describe the best I can of our situation for Operations. During the remainder of the flight the fuel indications appeared to showing various indications to the point I was concerned about how reliable they were. We ended up running the Fuel Leak QRH out of caution and continued to ZZZ2. We had an ALT; CF and ACF99 but we were eating into those fuels. I had a Dead Heading Pilot observe the left Engine and wing during running the checklist to confirm no visible fuel leak. He said he saw nothing unusual. After we parked at the Gate; Operations Personnel asked us what we had experienced. He said this aircraft had a history of this but it had not been written up in a month. That would explain why I saw nothing about this type of fuel problem but it did have a MEL XX-XX-X for a Number 1 Engine Valve Closed Light. The Operations Personal also said they wanted to take the airplane to the hanger to fix the fuel problem but were getting push back to keep it on the line. That made me mad and very concerned. I have never had to run the Fuel Leak checklist in the airplane and with no airfield immediately close by it was not a very comforting feeling. I had pages and pages of dents and scratches for the airplane but NOTHING about the very important history of the unreliable fuel system. I would definitely NOT fly this airplane again until properly fixed.