I haven't found direct stats on this, but given that according to the UN, in 2015 alone
Funding for malaria programmes has reached unprecedented levels. [...] Last year 88% of the 214 million cases [...] occurred in the WHO African region. [...] While more than 1 billion mosquito nets have been distributed in sub-Saharan Africa since 2000, there are doubts about whether people are using them properly.
So if you have 200 million cases per year, and these are prescribed a drug, in 5 years you'd get that billion prescriptions, although they might be the same individuals being prescribed... year after year. So YMMV. I've included the quote on nets distributed for comparison with another non-pharmaceutical anti-malaria intervention/prevention effort that has reached billions.
As hinted to me by @Taladris below, one also needs to consider that the malaria pathogen developed/develops resistance to drugs in this family. So there have been changes to the recommended treatment in various parts of the world, which make the claim less likely to be true. The following quote is from a 2002 source, and is mostly qualitative in nature, but should be indicative of the additional difficulty in evaluating the claim (about these two specific drugs), in the absence of precise statistics about them in particular/isolation:
For example, CQ is still the recommended first-line treatment for P. falciparum in much of Africa, despite the high prevalence of CRPF [Chloroquine-resistant P. falciparum]. [...]
Drug resistance is not an all-or-nothing phenomenon. In any given area, a wide range of parasitological responses can be found, from complete sensitivity to high-level resistance (see Table 3-4). In general, malaria parasites in western sub-Saharan Africa are less resistant to drugs like CQ and SP than malaria parasites in eastern or southern Africa. [...]
The policy response to increasing evidence of antimalarial drug resistance has been variable as well. In parts of East Africa, parasitological resistance to CQ is very high, with 80 to 90 percent of P. falciparum infections being moderately to highly resistant (Bloland et al., 1993). In response to these high rates of resistance, Malawi switched from CQ to SP for first-line therapy for P. falciparum in 1993. A number of countries in eastern and southern Africa (including Tanzania, Kenya, Democratic Republic of Congo, Rwanda, Uganda, Ethiopia) have made similar policy changes to SP alone or in combination (with either CQ or amodiaquine) on a national or provincial/district level. After a long period of disinclination to change treatment policies, many more countries in sub-Saharan Africa are now reevaluating their national treatment guidelines and considering policy changes to locally effective regimens. Although the drugs being used differ, similar efforts are under way in the Amazon region and Southeast Asia.