Some countries require that you have a card that says you can buy booze. Other countries will sell booze to people who look like foreigners. In Aceh, Indonesia, I knew non-Indonesian Muslims who got lectures from Sharia police for immoral behavior (mostly women who didn’t dress in a way the Sharia police liked) or were refused by wait staff to be served.
To get a card enabling you to buy alcohol at the state-run liquor store in Qatar, you need a letter from your employer stating your religion (and, as noted above, your monthly income, since your buying limit is based on your income). I’ve heard that bars are a lot less picky about whom they will serve. I’ve certainly never seen a hotel bar or restaurant ask anyone for an ID or anything. I dunno about converts.