Is it true? No.

Jesus was a Nazarene—from Nazareth, not a Nazirite (a particular vow of Judaism that did not allow a man to even eat grapes or raisins—so no wine, not even Welch’s grape juice, or cut his hair). Jesus not only drank wine, but he also created a whole lot of wine at the wedding in Cana, after everyone had already had a lot to drink.

At that time Jesus came from Nazareth in Galilee and was baptized by John in the Jordan. Mark 1:9

The Lord said to Moses, “Speak to the Israelites and say to them: ‘If a man or woman wants to make a special vow, a vow of dedication to the Lord as a Nazirite,they must abstain from wine and other fermented drink and must not drink vinegar made from wine or other fermented drink. They must not drink grape juice or eat grapes or raisins. As long as they remain under their Nazirite vow, they must not eat anything that comes from the grapevine, not even the seeds or skins.

‘During the entire period of their Nazirite vow, no razor may be used on their head. They must be holy until the period of their dedication to the Lord is over; they must let their hair grow long.” Numbers 6:1-5 and on through verse 21.

The giving of the instructions for those taking a Nazarite vow includes not being near a dead body; Jesus was often near dead people though they didn’t stay that way. If he’d been a Nazarite, he would have had to shave his head every time.