The term “GMO” typically refers to crops or animals that, through genetic engineering, have had a gene (or a few genes) from a different species inserted into their genome. So yes, by design, to improve a crop or animal with genetic engineering, the genome of the new, GE variety has been changed by the addition of new genes(s).
Your question also asks about whether inserting the new gene(s) will “…activate genes…” Some traits in GE crops, like drought tolerance, involve inserting a gene that, again by design, enable the plant to thrive when water is limited. For more information on GE crops like drought tolerant corn, check out the DroughtGard® product page and Innovations webpage on Monsanto.com. But for most GE crops, the gene introduced (e.g., to help protect the plant from insects or herbicide damage) does not affect any other genes in the plant. For a database of GM crop approvals, please review the Center for Environmental Risk Assessment (CERA) website.
All GE crops undergo an extensive molecular characterization to eliminate any plants in which the inserted gene(s) have disrupted other genes in the genome. A recent publication that gives more details on the plant selection practices used for GE crops is Glenn et al, 2017. And more detailed descriptions of methods used to make GE crops have been previous posted to GMOA and can be found here and here.
Your question suggests, however, that you are under the impression that genetically engineered organisms (which you call GMOs) have more possible genomic changes than would otherwise be possible. Methodologies such as southern blotting and whole genome sequencing have revealed that genetic engineering results in relatively modest changes compared to the other ways in which humans have been modifying the genome of plants and animals over the past many millennia for our benefit (such as for agriculture). Virtually all of the sources of food in our diets are from “genetically modified” organisms,” where this term more broadly refers to any plant or animal that has undergone human-mediated breeding and domestication for our benefit. Genetic engineering is a recently developed technology that adds a new method of modifying plants for human benefit. Ancient farmers “modified” plants when they developed crops, like wheat, rice, potatoes and maize. For example, wheat was created by humans breeding three different species that, if left to nature, would not have cross-bred. Fortunately, these farmers figured out how to make wheat since we enjoy the many tasty products from wheat, like French bread and Italian pasta.
Genome sequencing has also shown that conventional breeding can significantly affect the genome of plants. One example is the recent discovery that over 1,800 genes in grapes used for certain types of wine (Pinot Noir, Corvina, Tannat) are not even found in other wine grapes here is a recent publication on this topic. By comparison, GE varieties of maize have only one (or at most a handful) of new genes, out of the ~32,000 genes in the maize genome. To provide context for the ratio of one gene in 32,000, this is similar to a half teaspoon of liquid added to a full bathtub of water – not much. Mother Nature, not to be outdone by humans, is also a rich source of “genetically modified organisms.” For example, long before plant scientists discovered that the soil bacterium, Agrobacterium, can be used to insert genes into plants (the main method to make GE crops), natural processes had already resulted in genes from Agrobacterium being inserted into plants, like sweet potatoes. You can read more in this publication Kyndt et al, 2015.
To summarize, plant and animal genomes are constantly undergoing changes that activate, inactivate, insert or delete genes. Humans have used this genomic diversity to domesticate many plants and animals for food production. The genomic changes in GE crops (aka GMOs) pale by comparison to the extensive genomic changes that farmers have, and continue, to make through conventional breeding practices, or that Mother Nature uses to generate the wide diversity of life on earth.