I do not know why there is an error but not using Subscript it works :
DSolve[{q[t] + omc*q[t] - omc*q[t - 1] + 2*sig*q'[t] + q''[t] == 0,
q[t /; t < 0] == 0}, q[t], {t, 0, 2}]

Another Example:
DSolve[{q[t] + omc*q[t] - omc*q[t - 1] + 2*sig*q'[t] + q''[t] == 0,
q[t /; t < 0] == 1}, q[t], {t, 0, 1}]

Regards Mariusz.
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