Is this a Trapezium? I once read that in hyperbolic geometry, two hyperbolas can be parallel. In a trapezium, you have four sides and a pair of parallel lines, therefore is it possible to have a trapezium with two hyperbolas like so:

Is that a valid trapezium?
 A: In hyperbolic geometry, any line can have infinite "parallel" lines. So, it is possible to have a quadrilateral similar to that depicted in the figure, where the upper and lower sides are actually parallel hyperbolic lines.
A: STOP you are making to many steps in one go.
Hyperbolic geometry is not your well known Euclidean geometry and you need to explain every term.
Before you can even start to answer , you will first have to explain what you mean by a "hyperbole (on an hyperbolic plane)"


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*In hyperbolic geometry there is no general accepted cartesian coordinate system, or rather there are 4 or 5 of them.

*Also in eucledean geometry there are a couple equivalent definitions of what an hyperbole is , but in hyperbolic geometry they can result in different curves.
Then the sides of trapezium are line segments and why should an hyperbola be a line? or do you define a  "trapazium"(on an hyperbolic plane) as not consisting of (hyperbolic) line segments? 
Then in Hyperbolic geometry there is a difference between parallel and non-intersecting lines which of the two do you mean? 
(curiously in your drawing the "horizontal" lines are not (horo) parallel so are only non -intersecting or "ultra- parallel" , while the "vertical ones" could be anything.    
See it is not so easy to answer your question, there is much more YOU need to explain, before I can answer your question
Good luck
