B737 Captain reported diverting to an alternate airport after experiencing the apparent failure of an air data computer that resulted in multiple system anomalies.

Narrative

Lever cruise at FL360 we got a fuel imbalance message. I was PF and gave the AC to the FO and ran the checklist. As we were balancing the fuel the ride got bumpy so we got clearance to climb to FL380. In the climb the PFD (Primary Flight Display) indicated an airspeed disagree and altitude disagree. Since my instruments agreed with the standby info and while following the checklist I became PF again and the FO went through the rest of the checklist. As we were no longer RVSM capable we got cleared to descend. While descending we began to discuss our options to either continue to destination or divert. We got dispatch involved and weighed our options. In our decent to FL280 we saw both engines EEC (Electronic Engine Control) had gone into alternate mode. The FO ran the non normal for that. When I cruised the FMC for FL280 there were no more fuel quality indications only fuel total display in the FMC pages. We [advised ATC]. Briefed the flight attendant and let the passengers know we were diverting to ZZZ [airport] because of aircraft flight instrument issues. On arrival we also got a speed trim fail light and the FO ran the non normal. We landed ZZZ. The fire trucks followed us to the gate no other issues to report.