As a cold progresses, mucus often thickens and turns from clear to yellow or green. This colour change comes from immune cells responding to the virus and is a routine part of recovery, not proof of bacterial infection.
Antibiotics work only against bacteria and have no effect on the viruses that cause colds and most coughs. Taking them when they are not needed exposes you to side effects and contributes to antibiotic resistance.
What matters more than colour is the overall pattern: symptoms that worsen after improving, a high or returning fever, facial pain over the sinuses lasting beyond about 10 days, or feeling increasingly unwell. If those occur, see a doctor or pharmacist, who can decide whether antibiotics are appropriate.