We cannot know for certain either way. It’s certainly possible, though I tend to think it’s unlikely. We know that Germanicus was wildly popular with the people and Tiberius had never been at any point in his career. Tiberius had rebuked Germanicus for his military campaigns deep in Germania and then moved him to run the eastern provinces. It’s open to interpretation, but the general consensus is that Tiberius promoted Germanicus out of Germania to break his connection with the Rhine legions and to put an end to the raids, which he viewed as unnecessary.

As Germanicus was sent East, Claudian family-friend and known political rival Piso was sent East as well. He pretty much immediately started negating all of Germanicus’ orders and undermining him at every possible turn. It is extremely unlikely that he would try to undermine the crown prince without some form of political backing from either Tiberius or Livia. His behavior strongly suggests that either Tiberius or Livia was concerned by Germanicus’s growing popularity. That said, undermining Germanicus and killing Germanicus are completely different things. At this point, Germanicus got sick and eventually perished after a few weeks. Piso had already been sent away from Germanicus prior to his death, so if he had somehow been involved, he must have found an accomplice. After Germanicus’s death, Piso was recalled to Rome to stand trial.

The Romans themselves were left with many unanswered questions when Piso killed himself before his trial. Did he have explicit orders from Livia and Tiberius to kill the young prince? Was he told to undermine Germanicus to render him the politically impotent? Was he just operating on his own? Was his political rivalry completely unrelated to Germanicus’s death? Was Tiberius removing Germanicus so his biological son Drusus could be the sole heir?

We also have to consider the bias of our ancient sources, who uniformly hated Tiberius. Though Tiberius eventually devolved into a monster who’s paranoia got the best of him, at this point he was still actively and competently ruling. He was hand selected by the Augustus to rule the empire and, based off his military and political career, had complete stability in his position. He did not need to kill his adopted son to save his position in power, especially because Germanicus had shown nothing but deference to Tiberius.

Sadly we’ll never know, but it is fun to speculate.