Short DNA triplex strands have been identified in various organsims. But, would injecting a single strand of DNA or RNA (with its bases complementary to the bases of DNA found in the major groove of a certain gene ) cause it to form a DNA triplex at that gene?
Possibly, yes, but there are complications.
There are other conditions that help create the energetically-favorable conditions for bases to bind in a way to form a triplex, including acidity and presence of magnesium ions.
The region of DNA where your gene of interest lies (or its regulatory regions, depending on what you're trying to do) would need to be unpacked, as well. DNA is usually packaged into chromatin. Histone complexes, which wrap up DNA, provide some measure of protection of the major groove. To make a triplex, you'd need biochemical conditions that unwrap histones and support triplex formation, generally.
Use of RNA, specifically, could be complicated if it is or could be processed into a short- or microRNA element, which could be consumed in downregulating the population of mRNA molecules (which, in turn, ultimately regulates gene expression). It isn't helpful if it gets used up before it binds to your triplex of interest.
There is some pharmaceutical interest in TFOs (triplex-forming oligonucleotides) as a drug-delivery mechanism.
If some gene is broken and implicated in a genetic disease, say, and its DNA or nearby regulatory regions can be made accessible, perhaps a TFO can be administered that would help turn up or down the expression of that gene's protein product.
Papers on TFOs could probably point you to other engineering problems that researchers have encountered and would need to overcome in order to create an effective medicine.