Air carrier Captain reported during taxi an aircraft taxied in front of them requiring aggressive braking to avoid a collision. Captain reported that some flight attendants may have sustained injuries as a result of the sudden stop.

Narrative

Upon taxi out; we'd received instruction to taxi down [Taxiway] 1 to [Taxiway] 2 and hold short of 2. There was an aircraft on [Taxiway] 3 which was taxing faster than we were; it went right at [Taxiway] 4 in front of us and cut us off. We were not told to give way to any traffic; I did not expect an aircraft to cross in front of us left to right. I immediately depressed the brakes to avoid running into that aircraft. The stopping force was more aggressive than a normal taxi stop. We asked ATC what happened; she said standby and later apologized for not 'pointing out the traffic'. First Officer (FO) was my Line Check Pilot (LCP) whom just completed my line check with on flight before. He called the back to check on the Flight Attendants (FAs) and reported back to me that everybody was good to go. We continued taxi. 45 minutes later we took off. Upon reaching cruise altitude; the purser called to let me know that the aft FAs were claiming they were injured. I asked to speak with the aft FAs a couple times; only Flight Attendant A called me back. Flight Attendant A stated the 'cart almost ran her over' and XXX banged his hip. I asked her to elaborate on her injury since almost running her over wasn't clear to me what actually happened. She wasn't very clear but said the cart hit her foot. I asked if they needed assistance; they said no. They also told me they'd already written the incident up instead of calling us; that's apparently what they felt was the appropriate first step? I asked her if she heard us ask them on the interphone if they were ok; she said no. This was the first time I was being alerted of a possible interphone inop situation. I wrote it up upon landing. I asked Flight Attendant A if she heard our 'be seated' for departure announcement - she said yes. I asked her why she didn't call me at that time so we could address the injury; rather than taking off. She said 'we were trying to figure out why you didn't call us'. They refused the need for medical attention numerous times; and insisted on doing their full service. They did however want to go to the clinic after arrival to ZZZ1. Less detailed report was filed as per FOM. Chief pilot and Dispatch were called as well (per FOM).