SpaceX's Falcon 9 uses TEA-TEB as a ignitor for its Merlin engines. My question is if the rocket has more TEA-TEB, can the engine be shut down and restarted more than 4 times?
The engines can most definitely be ignited more than 4 times.
Every engine is separately test-fired at McGregor after it is built. The entire booster is then again test-fired at McGregor before being delivered.
In general, there will be a static fire of the fully integrated vehicle 1–2 days before launch.
So, even for a drone ship landing which has no boost back burn, the outer engines will have been ignited four times, the middle engines five times and the center engine six times.
And that is assuming the booster crashes on the first landing.
If the booster survives the first landing, the engines will be fired many more times; at least once for each launch.
Recently, SpaceX has launched Falcon9s without a static fire, so that reduces the number of ignitions somewhat. However, they always do a static fire on new boosters, and typically also on life leaders (i.e. on ones that are either not yet flight-proven or that are pushing the limits of reuse).
They might be swapping out engines when they see some anomalous readings, but we don't really know that. Unlike for Starships, this doesn't happen in plain view on the side of a public road, it happens inside the hangar.
However, even assuming that they have swapped some engines, it is likely that at least one of the engines on B1051 was ignited more than ten times.