Section 3.4 of the document that was linked by @mara. Take a few minutes and read that.
Yes and no. You have a model object that has no connection to the area under the ROC curve. If you want to get that, you need to make predictions and then calculate that.
The problem is that, if you just repredict the training set, the AUC will be inaccurate. Resampling is your best approach to estimating it correctly. I suggest taking a look at the caret package to do that (but please read the docs before asking questions about that).