I purchased a 220V washer/dryer, I have 110V currently. Can an electrician convert this? And how much should it cost?

No. This is NOT possible.

A typical electric dryer in the US is a 120/240v-30 amp circuit. You will need a new circuit run if one does not exist.

This could run $150, or $550, or more. Depends on the setting and your location.

Curious, why did you wait to ask this question until after the purchase? That dryer is going to cost you double now to get your new 240 in place.

Your house has 220 coming in. only one leg is used for 110. You will need a special circuit breaker to be installed in your box. If you do not know haw to do this.. I assume you do not.. a professional electrician can do this and run the wires for a plug to power your appliance. Really it should not cost too much and you may already be set up for it since that is common for most houses.

There are 120 volt dryers on the market. Of course, the capacity is smaller and they are also expensive.
If you need a fullsize dryer I don't see much you could have done differently.
I would try to find an electrician that does some work on his own. I know a few, but that doesn't help you. Electrical supply stores will sometimes give recommendations.