Broadly speaking, uncertainty happens when multiple input values, or causes to be inferred, are associated with the same outcome or result. In other words, for the same value of y, there are several values of x that might have caused it. Thus, when we observe y, we cannot (always) infer the value of x with perfect confidence - rather, a number of possible values of x are consistent with the observed y.
This can happen for a number of reasons, but they fall into these main categories: deterministic causes, random causes, and imperfect knowledge. A deterministic cause of uncertainty is if the (deterministic) function that maps x to y isn't monotonic. For instance, the function may plateau at some point, such that further increasing (or decreasing) x no longer changes y. Or, the function may go up and down and have one or more peaks and troughs, such that on either side of these peaks and troughs, and more generally throughout the function, we get identical values of y for different values of x. For example, if x maps to y via a Gaussian function, then for each value of y within the function's range, there are two values of x, at equal distances from the center of the Gaussian, that produce that same value of y (with just one exception at the center of the Gaussian). Thus, for (nearly) every value of y that you might observe, your uncertainty includes two possible values of x that might have produced that outcome.
Random causes of uncertainty include anything that you cannot explain deterministically in your model. True randomness is often referred to as "noise", but it's often hard to define what is really random. For instance, when comparing different groups of people on their body height, you might assume their heights vary randomly around some population mean. But is that variation really random? Probably it is more accurate to say that it is the sum of many causes that are impossible for us to measure or know about (or at least prohibitively difficult). This is how it typically goes. As far as we know, the universe is only really random at the quantum level, and most of the things we call random in practice are actually just chaotic or complicated. But that's fine, because we can still treat them as random for statistical purposes and apply the rules of probability to them.
So, the third cause of uncertainty is actually often the largest one, and it is imperfect knowledge. Only, in statistical models, such uncertainty usually gets absorbed into a random term, for the reason I stated above. Imperfect knowledge can range from "unknown unknowns", such as the many causes of height variation mentioned above, to known unknowns, which is when e.g. you know the model that generated your data, but you don't know some of the model parameters, or you know them only up to some margin of error.
In conclusion: yes, it is possible to have uncertainty without noise. First, it is possible if there is ambiguity in the deterministic mapping from causes to outcomes, via a non-monotonic function. Second, it is also possible when this mapping is deterministic and unambiguous, but we just don't know the function precisely. Finally, it is possible if you take a very narrow definition of "noise", to mean only truly random variation, but I assume that isn't what you meant.