The short answer is yes, there are, in some cases, although none of them is situated in Great Britain or Ireland. Of course, horse racing is popular in many jurisdictions globally, so your correspondent will leave it to the learned readership to say otherwise, but it appears that there are no racecourses beginning with Q or X anywhere in the world.
As far as the other letters are concerned, rather unhelpfully, the Italian and Russian words for ‘racecourse’ are ‘ippodromo’ and ‘ippodrom’, respectively, both of which, of course, begin with I. More specifically, though, Inverness Raceway is a harness racing track on Cape Breton Island in Nova Scotia, Canada, while the Inner Mongolia Racecourse in Hohhot, China is the largest racecourse in Asia, for all that gambling on horse racing has been illegal on mainland China since 1949. The letter J has a few more takers, worldwide, including Jydsk Væddeløbsbane or, in English, ‘Jutland Racecourse’, in Aarhus, Denmark, Jokimaa, in Lahti, Finland, Jarlsberg Travbane in Tønsberg, Norway, Jägersro Galopp in Malmö, Sweden and Jebel Ali Racecourse in Jebel Ali, United Arab Emirates, to name a handful.
Racecourses beginning with V are even more plentiful, it seems. The Hippodrome de Vincennes, in Vincennes, in the eastern suburbs of Paris, Hippodrome de Viseo, in Zonza, on the island of Corsica, and Hippodrome des Vignetta in Ajaccio, also on Corsica, are just three of those on French soil. Further afield, Hipodromo de Valencia, in Valenica, Venezuela, is one of three major horse racing venues in the country, while Valparaiso Sporting in Viña del Mar, Chile is the home of the Chilean Derby. Racecourses beginning with Z, though, seem to be a real rarity; Zagrebački Hipodrom, or ‘Zagreb Hippodrome’, in the Croatian capital, appears to be the sole example.