Calculating charge given current and capacitance over time
The direct answer to your question is no.

The missing piece of information is the initial voltage (or initial charge) as you correctly indicated in your formula for V.

Think about filling up a bucket with water. Your charge is the volume of water, your current is the flow of water, your voltage is the water level inside the bucket and the capacitance is a function of the geometry of the bucket (the bigger the area of the bucket, the bigger the C). If you don't know how much water you had in the beginning, you can't tell how much you'll have in the end.