Imagine a plane that's infinitely large. Would there still be a center of it? I feel like there couldn't be one... But something tells me there should be in some way!
Your title allows for better answers than your example. And I will answer from the title question, rather than limiting myself to the plane.
You know where the centerpoint of a parabola is, even though it extends infinitely far away from there in both directions -- it is the point closest to the focus, the point of maximal curvature. It is a center in a specially defined sense which recognises a sense of the symmetry a center should confer. Similarly a cubic has the opposite kind of center, a uniquely flat point. However you rotate the curves, these central references remain obvious, and they are historically relevant enough that the ordinary sense of center is often extended to include them.
A hyperbolic paraboloid has a saddle point where its tangent space splits it into ascending and descending parts, and that makes it a centerpoint of the space in important ways, again related to symmetry rather than distance.
So in the sense of the point that everything is intuitively situated in terms of, like a city center, certain infinite spaces have centers. If our own universe is "too light", it may have a hyperhyperboloid shape, with the third-dimensional equivalent of the saddle point, and in this sense a relative center.
As implied by Conifold and RodolfoAP the answer depends both on the definition of centre and the nature of the space.
If we choose "centre of symmetry" as the definition then (for example) an infinite plane can have either one or an infinite number of centres, depending on its structure. The example of a series of uniform concentric rings implies a single centre. If there are identical structures at every point (m,n) (m and n both integers), there will be a centre of symmetry at every point (r/2, s/2) (r and s both integers). If the plain is continuous and of uniform density, then every (finite) point on the plain will be a centre of symmetry.
Various writers cite the universe. It is generally believed not to be repetitively cyclic, in which case it will be infinite in whatever number of dimensions is needed to allow it to be defined as a stationary entity (this would ignore the possibility of a universe corresponding to every possible outcome in the present one, which would seem to map to an infinite number of dimensions). QM appears to preclude a regularly repetitive universe, which means that our universe would appear to have a single centre, albeit the location could be uncertain to a Planck length or so.
The definition of center is your issue. This is not philosophy or math, this is a rhetoric problem.
If center means the axis of rotation and your plane rotates, there is your center. Centers are equidistant points to sets of systems (borders, lines, weights, moving points, etc.). Then it depends.
This is like asking if you can think of nothing. It all depends on your definition of thinking. There are no absolute truths on language, where concepts are usually subjective. Only on logic and causality (except that causality has been broken on Q. Physics and we're rebuilding our concepts... Again).
the concept of infinity in mathematics, means undefine. can you define something on top of undefine? no, at least mathematically.
however, on top of something which is not only mathematics, and it is a progressing infinite object, like an object growing on the forever looping computer program, yes, you can.
for example: assuming you have defined a point as 0,0, and a program loop forever to extend both of the axis to infinity. you will achieve infinity plane, and 0,0 is your center. the same concept applied for any dimensions and thus explained any objects in the universe.
however, for something already created, and especially you are within its lower dimension, you can never find the center. as you don't even able to find the whole picture of it, for you, as civilized human, inifinity, is just undefine.