no,no.
[One might wish to argue today's meter IS different from tomorrow's because the time dimension, the fourth dimension, is different...but the length is the same.]
The universe is 'expanding' only over intergalactic distances...on Earth and in our solar system for example, gravity holds everything together. [I guess it might vary imperceptibly over time as masses [suns,planets,etc] change positions...but is way finer than our ability
to observe (measure). For example, if a gravitational waves passes, curvature varies and so would distance.] Space and time are fixed and immutable.
The usual 'fourth dimension' is time. I don't think anyone knows for sure what our universe of spacetime is expanding into, perhaps 'nothing', perhaps what existed before the big bang.
We don't know for sure if the universe is infinite or finite, bounded or unbounded. A cyclic theory has us and our dimensions endlessly oscillating between branes which provide boundaries.
In another sense the question about what we are expanding 'into' is like asking, what are we expanding 'from'?? Usually when you 'leave one spot' you 'enter a new spot'...not necessarily so in cosmology.