Can I say that I draw up accounts instead of doing invoicing?

I don't think so-- to draw up accounts means to me to create them.
"drawing up accounts' really describes the annual accounting activity of a company - the sort of thing they have to submit to the regulatory bodies in their country (for example, Companies House and HMRC in the UK). It doesn't normally relate to the company's client accounts directly. An accountant normally draws up your accounts - showing your trial balance, bank reconciliation, financial statement etc
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When there is an invoice I just fill it out with my data and with the data of my client and it means that I make out an invoice or I do invoicing.

I thought that it's maybe the same with an account - it's something a bit different than an invoice. I need to fill it out appropriately, just like an invoice and send it to my client, and then, I guess, we say that it is drawing up an account.
Well, if that's what you say, then use it. I merely gave you my opinion.
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 nona the brit's reply was promoted to an answer.
OK. Thanks for the replies.