Do I need to add a vent when moving a toilet drain across the bathroom?
You must have a vent. The toilet will not flush properly without one, and there will rarely be enough water left in the bowl. You need a 2" vent to take off from the branch line within 6 feet of the toilet. It can run up any convenient wall until it is above the toilet's flood level. Then it can run near horizontally (1/4" per foot) anywhere to tie into an equal or larger vent.

There's any number of ways to properly vent a toilet, and several common ways that are technically wrong, and even more that are just plain wrong. It's impossible to suggest a good method without a lot more information.