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Since you do have a viral infection already in the form of common cold, it does look like the redness of the eyes is likely to be due to viral conjunctivitis.
Viral conjunctivitis is one of the causes of pink eye and is usually self limiting.
Its natural course is to subside by 1-3 weeks without any specific treatment.
Lubricating eye drops can by used for symptomatic relief. You can try using preservative free Refresh tears eye drops, 1 drop 4 times a day, for 1- 2 weeks.
This eye drop is available OTC.
You can wait for the signs and symptoms to subside over the next 7-10 days.
If the symptoms persist or worsen, then do get the eyes examined by an ophthalmologist for a closer look .
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Views expressed are for information purpose only and cannot substitute a visit to an ophthalmologist