"In principle," if you worked for two days in a state that did have income tax you would have to check the filing requirements for that state. Each state has different rules for who is required to file a state tax return. The amount that you earn in two days might not be enough to be required to file. But in some states having any income at all from that state might require you to file. It wouldn't matter where in the state you worked. Whether it was in the company's office, a conference center, your hotel, or Starbucks, you worked somewhere in that state.
That's "in principle." In reality, most people would ignore a two-day business trip to another state, even if, according to the letter of the law, it required filing a tax return in that state. That's assuming that the two-day trip was your only presence in that state for the year. But if, for example, you spent two weeks working in another state, or you made a lot of two-day trips, that would be harder to ignore.
