A320 pilot reported airframe vibration resulted in a divert with expeditious handling to a safe landing.

Narrative

During climb out from ZZZZ; we started experiencing airframe vibrations at approximately 280 knots at FL200. We slowed down to 260 knots; approximately Mach 0.75; it started to improve. We attempted to increase speed and the vibration would come back immediately. When we called the inflight; they informed us they felt a very strong vibration in the back; specifically near row XX and beyond. Once at FL340 we still felt some vibration at .75. We initiated a satcom call with Dispatch and Maintenance. We attempted to gently increase the speed and it immediately got much worse. At the mean time we got informed by inflight that customers at row XX were getting very uncomfortable and some were starting to cry.Since the vibration wasn't getting any better and after agreement between us (CA and FO) Dispatch and Maintenance we decided that the best course of action was to divert to ZZZZ1. We informed the inflight and customers of our intentions; and diverted to ZZZZ1; performed the diversion and overweight landing checklist. Landed approximately 5000 pounds overweight and less than 100 FPM descent at touchdown. After arrival made multiple logbook entries; and fill a report.