What is the quickest way to gain morale in a conquered province.

Spending gold is by far the fastest way to increase province morale. Otherwise, you must wait for day change for morale up update. You can increase how much morale provinces gain at the next day change, by building fortifications. Placing a garrison can help to lower the province's chance to revolt due to low morale.

Conquering enemy capitals will also provide an immediate boost to morale to all of your provinces.

Righto, cheers!