As written, none of this applies to the quickling, because they simply have a very high natural speed and are not flying, or powered by magic, or anything like that. However, these rules are the only alternative the game offers to the normal rate of travel described in the PHB, so if you rule that quicklings should count as having a special travel pace, this is what we have to work with.
Using these rules, a quickling travelling at fast pace for 8 hours a day would cover 1500 miles in just under 12 days. (Note that this is very close, but not identical, to simply multiplying the standard rates of travel described in the PHB by 4, which is congruent with the assumption of a 30ft standard movement speed in those rules, as used in Thomas Markov's answer.)
Quickling Dl
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With a +1 constitution save modifier, they will on average be able to travel for an extra two hours before one level of exhaustion sets in, from which they recover while they rest; a quickling making such haste would average 160 miles per day at a fast pace, reducing a 1500-mile journey to about 9 and a half days.
Just for the sake of completeness, a quickling that was immune to exhaustion and did not need to sleep could max out their speed at 384 miles over 24 hours, and cover a 1500-mile distance in almost 4 days. Unfortunately, despite their description as "never truly at rest", nothing in the quickling's statistics actually prevent it from needing to sleep or becoming exhausted.
If you apply the same principle to a quickling postal network, and allow that there is a network of rest stations for quicklings such that one courier can do their full day's travel and then pass on their deliveries to another quickling courier, a Quickling Express delivery could be moving at very nearly 24 hours per day and travelling nearly 384 miles per day (quicklings have darkvision, so there's no reason they can't travel at night). This requires such a rest station roughly every 160 miles along the journey's route, which doesn't seem an unreasonable prospect for an organised postal service.
Quicklings are not mounted nor mounts, but if you decided to apply this rule to them as well, a quickling at a gallop could cover 32 miles in one hour, and if a sufficiently dense network of Quickling Express stations existed that the delivery could be handed off to a fresh quickling every hour, the delivery could go at nearly 768 miles per day, reducing that 1500-mile journey to just under 2 days. Of course, that does depend on the transport network having a Quickling Express station every 32 miles and a very large number of cooperative quicklings, the former which may be infeasible and the latter of which seems very unlikely.
PCs usually have a speed of 30 feet, and the numbers on that table are written with that assumption. Since your quickling courier has a base speed of 120, which is four times 30, just use the base measurements in the travel pace table multiplied by a factor of four.
A PC with a speed of 30 feet can travel 24 miles a day at a normal pace, so a quickling should be able to travel 96 miles per day at the normal pace. At a fast pace, a quickling can cover 120 miles per day, so should be able to cover your 1500 mile target in 12.5 days.
Do you want your postal service to be faster than normal mundane speeds? A Quickling have twice the speed of a regular horse, and although some can argue that he could get tired faster, for example, there's no reason not to use the same mongolian system in a postal service (a bunch of outposts with rested quicklings ready to resume the delivery). This makes it possible to have a non stop 24h service, bypassing the standard barrier of how long a single creature can travel per day). 2ff7e9595c
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