Is the diameter of the observable universe a relative quantity? The diameter of my observable universe is 90 billion ly measured in proper distance. But isn't lenght a relative quantity in the theory of Relativity? Could an observer moving at a different velocity than me measure a different diameter for my observable Universe?
If the observer has always been moving at a high speed, then yes, s/he will measure a different value than you. But if s/he accelerated to a high speed just now, you would obtain the same result.

The reason is that the radius of the observable Universe is defined as the distance light has had the time to travel since the Big Bang, and that does not depend on your instantaneous velocity.

However, if an observer started moving at, say, 0.866 times the speed of light when the Universe was born, then in the reference frame of that observer, the Universe would only be $\sqrt{1 - 0.866^2} = $ 1/2 the age measured by you, i.e. 6.9 Gyr. At this age, the radius of the observable Universe was only ~39 Gly, i.e. ~0.8 times the present value.

Note that in order for this result to be exact, the observer would have to slowly accelerate. The reason is that in an expanding Universe, a given "peculiar" velocity wrt. the surrounding matter will slowly decrease, because the comoving coordinate system expands with the Universe. For instance, an observer starting out with $v = 0.866c$ at the time the cosmic microwave background was released 380,000 yr after the Big Bang, would only be moving at 80 km/s today if s/he didn't keep accelerating.

If the observer were clever, s/he would probably realize that s/he did't live in an asymmetric universe where all matter seemed to be whirling by at close to the speed of light, but rather that the Universe were quite isotropic, but s/he had somehow acquired a high speed with repect to everything else. It would then be possible to deduce the "true" age of the Universe.