No, typically you can't ask for a raise within a week. In fact, usually asking for a raise within the first several months is considered a no-no.
Generally, the only exceptions to that rule would be:
1) If you're promoted
- Promotions typically come with increased compensation.
2) If you're given significantly increased responsibilities from when you first started
- This can easily lead to the conversation of "with the new responsibility, does it make sense to adjust my compensation?"
3) If the job otherwise changes significantly (such as a 40-hours-per-week job becoming a 60-hours-per-week job, or changing from a day shift to a night shift)
- Here I'm grouping lots of changes within the company where they may just expect you to understand "this is how things will be now", and not offer compensation as a result. You can still raise the point, but I would caution you not to expect a positive outcome.
4) If the company has a standard raise period (usually around the end of the year)
- Some companies have rules about how long you've had to be with the company before you're included in the raise conversation; often if you've been there less than 3-6 months or so, you'll be excluded. You can ask, especially if you're just outside the window; depending on the company, the rules may be more or less strict.
None of these apply to you - you're essentially feeling "buyer's remorse" at having accepted a position that turns out not to be as great a deal as you originally thought. From the company's perspective, nothing has changed from last week to this week, so a discussion about "can I get an increase in pay" will seem to come from nowhere, and it's likely the company will both turn you down and gain a negative reputation of you as a result.