I would suggest to call the clinic immediately, and go back for a follow-up, if you have already left the clinic. This is because a blue-ish discolouration could mean a few different things, and really needs to be assessed in person, to determine what the blue-like appearance is.
Firstly, it could potentially be a bruise forming under the surface of the skin. Bruising is common after injectables, as needles entering the skin have the potential to hit blood vessels. Bruising may not happen immediately after injection, and may occur hours after injection.
Secondly, it could be something called “Tyndall effect”. Tyndall effect usually happens immediately after dermal filler injections. Dermal filler that is placed superficially under the skin has the potential to cause a blue-ish appearance. This is the way the light reflects off the skin where the filler is. This is often caused in a tear trough, where the skin is thin, but can also occur in the nasolabial folds, the marionette area, lips and peri-oral. The only way to get rid of this is to dissolve the filler.
Thirdly, and worst possible case, is that dermal filler is a gel-like substance and has the potential to block an artery or a vein, resulting in restricted, or complete obstructed blood flow to a certain area in the face. This can occur in another area compared to where the filler was actually injected. If the skin is looking dark blue/ black in appearance, this is because there is no oxygen in the blood to supply the tissue and the end result is tissue necrosis (tissue death). This is very serious and you must present to the clinic immediately, so the filler can be dissolved and the tissue to become oxygenated with blood supply, again. If this is left, it can lead to permanent scarring and deformity.
So, if you notice anything blue-ish after dermal filler, always best to contact the clinic immediately for a follow-up. Better to be safe, even if it turns out to just be a bruise, which is temporary.