There are three ways to become a knight in Game of Thrones.
Another knight can make you a knight:
Tormund: I'm no king. But if I were, I'd knight you 10 times over.
Jaime: You don't need a king. Any knight can make another knight.
Game of Thrones, Season 8 Episode 2, "A Knight of the Seven Kingdoms"
As Podrick was Brienne's squire and served under her for quite a long time, it is very possible that she knighted him:
Podrick beams next to Jaime. Brienne paces. Bronn stands in the background.
Brienne: I don't need a squire.
Jaime: Of course you do.
Game of Thrones, Season 5 Episode 4, "Oathkeeper"
A king can also make someone a knight, as King Joffrey Baratheon did to Ser Bronn because of his role in the Battle of Blackwater:
Podrick: Ser Bronn of the Blackwater was anointed by the king himself.
Game of Thrones, Season 3 Episode 1, "Valar Dohaeris"
As Podrick appears to be a member of the Kingsguard, it seems possible that Bran could have anointed him for the position on the recommendation of Brienne and Tyrion.
Lastly, we know Lords and Ladies can knight someone, as Stannis did to Ser Davos for smuggling the onions into Storm's End under the siege. It's possible that as Podrick was Tyrion's squire for a while and Tyrion was quite fond of him he also could have knighted him.
I'd lean towards Brienne having done it, but lacking further evidence, we don't know for sure.