Can the system remind my staff to follow up on quotes my customers never replied to instead of us trying to remember?

Yes. A travel CRM treats a quote as an open item rather than a sent email, so every quote that has not turned into a booking stays on a list with a follow up date against it. When that date arrives, the quote shows up on the task list of whoever owns it, so nothing depends on someone remembering a conversation from last week.

In practice you decide the rhythm you want. You can set the system to raise a reminder a set number of days after a quote goes out, then again later if there is still no reply, and you can have it send a polite message to the customer at the same time with the quote attached or linked. Many setups also show you whether the customer opened the quote or clicked through to it, which tells your staff who is worth a phone call and who has gone quiet for good. As soon as the customer books, or the quote is marked as lost, the chasing stops on its own.

There are honest limits. This only covers enquiries that were actually entered into the system, so if your staff still quote from their own inbox or write phone enquiries on a pad, those quotes will never be chased. Automatic emails also depend on your sending setup and on the customer's spam filter, so some will simply not be seen, and whether text or messaging app reminders are available depends on your plan and on the messaging rules in your country. The system can nudge at the right moment, but it cannot do the persuading for you.

It is worth going through your recent quotes and counting how many ever received a second contact, because that number usually decides how much a follow up routine is worth to you. Then agree the timing and who owns an unanswered quote before you switch any reminders on.