A simple queue can just be an array in which you add new items at the end of the array with the push operator, and remove oldest elements from the front with the shift operator (you could also use pop and unshift, but it seems less natural and it is probably slower).

For a "two dimensional queue", although your description seems to be far from clear in my eyes at least (you don't specify how the items should be managed), you could in principle use an array of arrays (AoA). Or maybe what you really need is just several queues working in parallel (with different types of items). Or, if you really think of a queue within the queue, then you should probably just flatten the whole thing into a single queue, which will work just the same way.

But the very idea of a two dimensional queue seems to somewhat defeat or contradict the essential idea of a queue, i.e. a FIFO data structure in which the oldest element is removed first.

This tends to sound as a case of XY Problem. Please state what you are actually trying to do, rather than asking how to implement what you think is the solution to your problem.