Oooh that is a fun question! Maybe a bit about my Gram headcanons first before I get into his and Whitebeards relationship.

Polo Gram

Gram and Marco are very similar personality wise, laidback and very social

That being said, Gram is considerably more extroverted than Marco and can manage to get a conversation with just about anyone going. Whenever his ship docks anywhere, his first trip on land will take him to a bar or around the harbor to chat with the locals

I think Gram is a very 'do not more than is necessary' person while Marco can go above and beyond (and ignores his own limits sometimes)

Gram acts with deliberation, always weighing out the odds and what he and his crew can gain. Instead of bothering to fight through Marines or other Pirates to grab some of the treasure on God Valley, he waited until the Rocks Pirates were busy fighting amongst themselves to steal some of their treasure for his own crew.

Gram and Whitebeard (Not all of these are ship headcanons)

Whitebeard puts his drinks on Gram's tab, Gram does not mind as long as it means that they will go grab a drink together

Whitebeard is the type of person to speak his mind and intentions very clearly - Gram always find this "bluntness" of him amusing (if not a bit endearing)

He has to be pushed to admit it, but Whitebeard does find some of the jokes Gram makes very funny

Gram likes to lean against Whitebeard whenever he is tired, drop his head on his shoulders or drap himself over him whenever Whitebeard is sitting down.

Whenever Whitebeards bounty gets raised, Gram sets out to get his hands on his updated bounty poster. He likes showing them off too. Whitebeard just huffs at his antics.

Whitebeard complains that Gram is too chatty, Gram will accuse him of being anti-social and always grumbling into his beard

Similarly Gram will always complain about Whitebeard taste in alcohol being so cheap while Whitebeard only shakes his head at Grams more expensive taste sometimes

Whitebeard is happy living a simple life, he budgets his share of any treasure and always sends money back home on the dot. Gram is not irresponsible with money, but he likes to indulge every now and then

(Whitebeard is the 'we have food at home' parent, Gram is the 'sure lets eat out' parent)

Gram sometimes teases Whitebeard by challenging him to a fight. Whitebeard indulges him once (which Gram looses) and always turns him down afterwards