The temperament on some ukuleles is not perfectly equal.
I don't know if it depends on strings or on the instrument itself, but it means that in some cases (at some frequencies) the distance between two neighbor frets differs from semitone.
Moreover, these distances can also somehow depend on the current frequency of open string -
that is, the different overtones can be presented in various cases.
It can be heard especially when compare C-string with the others.
To prevent this, never tune your C-string, held on the 4-th fret, to E-string.
This will cause a dissonance between C and E open strings.
The simplest way is to tune all strings with help of tuner.
If you don't have one, you can download the APTuner program (for Windows).
However, if you can hear some intervals besides unison, the result of tuning open C-string to any other should be much better as well.
As for myself, I use the third fret of A-string and so tune C-string with octave.
music.stackexchange.com/users/23035/yaeuge
I can't speak to your particular model of ukulele, but I'm going to hazard a guess that your problem is similar to one that I've had in the past.
On my ukulele (a Fluke), there are small screws in the ends of the tuning pins that control the tension of the tuning pin.
If this screw is too loose, then the tension of the string will eventually unroll the pin, and the string will loose tension, causing it to go flat.
I've even seen it to the point where the pin is incapable of holding any tension whatsoever, and the string just sits there flopping loosely.
The solution is rather simple: find a small screwdriver, and gently torque the screw until the pin is able to hold a tension.
Be careful here: you aren't assembling furniture, and the screw does not need to go in as tightly as you can make it.
A quarter turn might even be enough.
If you overtighten it, it will be difficult to turn the tuning pin -- do not force it!
Just loosen the screw a bit, and you'll be fine.
You have to find a balance for the screw, between being too tight and too loose.
You can just barely see the screws in the following image, as small silver dots at the outside edge of the black tuners:
music.stackexchange.com/users/10637/Caleb Hines
I can't find your model, so don't know how relevant this is.
I have had various ukes of varying price, and an inability to be tuned correctly across all strings seems to be a common factor of the cheap ones; or at least a highly variable ability; I presume because they don't bother checking machine head quality or intonation but just knock them out at speed.
If you got it from a shop it is probably worth taking it back and getting them to either demonstrate that it can be set up correctly, or to replace it - even if it's cheap it should be tunable or it is not fit for purpose.
Another factor is possibly the strings; I have before accidentally tuned a string far too sharp on a ukulele, and it was completely impossible to tune that single string afterwards; it needed to be replaced.
This can happen if you are using the strings it came with if the uke was knocking around a shop for a while being abused by customers.
I also bought a cheap uke for a godson that had an appallingly stretched string out of the box, so clearly something went wrong at the factory.
So, to answer your question, no, it shouldn't happen, although it does on cheap ones.
If it's the strings, a new set will sort you out.
If it's the machine head slipping or the intonation, you'll have to return it.
music.stackexchange.com/users/18292/Whelkaholism
The string might be sticking in the nut slot.
I've found a bit of chapstick in the nut slot can free things up.
Or maybe the slot needs to be widened a bit.
music.stackexchange.com/users/21506/Steve Wellens
A few possibilities come to mind If the string is new or never been tuned up to pitch before then it might not have bedded in yet - nylon strings can take a good few days to settle down.
The string might not be tied on properly and be slipping against the tuning peg.
The string is faulty
The tuning peg is faulty (or badly adjusted as per Celeb Hines answer) If it doesn't settle down after a reasonable bedding in period I would take it back to the shop.
Also can't help but notice that it's the C string that's giving trouble.
This is the ukulele equivalent of the guitar G string which is notorious for being the hardest string to get in tune correctly.
much has been written on the subject.
music.stackexchange.com/users/20215/Noel Walters
I'm having some troubles trying to tune the C string on my ukulele (the biggest of all the 4). I happen to have no issues with the other ones, but this takes so much time to to be tuned and it won't stay in tune for a lot. Is it normal?  My ukulele is a 30E, Mahalo, and it has been my 3rd day with it. All the other strings seem to have settled, but the C string is so annoying.
difficulties tuning C string on ukulele
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<guitar><strings><string-instruments><ukulele>
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music.stackexchange.com/users/23032/Massimo Pesavento
Inability to tune all the strings is a common problem found on cheap ukeles. It is generally agreed that, whilst the C string should easily detune, it is a lot more common on cheaper ukuleles. 
It is generally agreed that, whilst the C string should easily detune, it is a lot more common on cheaper ukuleles. 