No. That's entirely a myth. There haven't been any snakes in Ireland since before the ice age. Almost everything we think we know about Saint Patrick is a myth too.
In the year 596, Pope Gregory sent emissaries to the British Isles for the purpose of bringing the existing Celtic Church under the authority of Rome. That was a century after Patrick's death, so we can deduce the form of Christianity that Patrick practised from reports of what the expedition from Rome discovered.
The Celtic Church had been founded by original Apostles, such as John, Paul, and Philip, or their immediate followers, and had continued following their original biblical religion for centuries. Gregory's Roman emissaries encountered a church whose teaching was very similar to that of the first century church, a church that was very different from the church in Rome.
The Celtic Christians followed the original teachings of Jesus and the Apostles. They honoured the Sabbath on the seventh day of the week and celebrated the festivals and holy days prescribed by the Bible. They baptized using full immersion, and abstained from unclean meat.
The Celtic Church rejected entirely any doctrines added by Rome. Immortal souls, an afterlife of bliss in Heaven or of torment in Hell, the doctrines of the Trinity and transubstantiation, the veneration of Saints or worship of Mary, and the use of statues, relics, and other forms of idolatry would all have been seen as pagan practices, unacceptable to a Christian way of life.
Similarly, they rejected pagan holidays that had been given a superficial Christianized renaming, such as Christmas and Easter. They did not accept the primacy of Peter. The word of God, through the Bible and as practised and taught by the original disciples, was their authority, not a Roman Pope claiming to receive infallible information directly from God.
Almost everything you know about Saint Patrick is wrong. He couldn't have used shamrocks to help teach the Trinity, as that was a doctrine he would never have believed himself. He wouldn't have condoned the existence of any saint's day, much less a Saint Patrick's Day.
And to top it off, Patrick wasn't even Irish, he was Scottish.