Are intergenic sequences and introns considered genes?

The better question is: Are there genes within intergenic sequences or introns?

And the answer depends on whether you use a modern definition of “gene” or the traditional one.

The traditional definition is a gene is a sequence of DNA that codes for a protein or polypeptide, with mRNA as the intermediary. And that’s still a pretty solid definition for prokaryotic genes.

Now that we know a lot more about how a eukaryotic organism’s genome functions, the definition of a gene has necessarily expanded. A common definition in molecular biology is that a gene is DNA sequence that codes for something functional.

Something functional certainly includes polypeptides and proteins, but also includes DNA sequences, which function as an “enhancers” or “silencers” of expression. Many of these are found within intronic and/or intergenic sequences. They can even be kilo-bases distant from the gene being regulated.

The definition should also includes DNA which is transcribed to regulatory RNA such as siRNAs & lncRNAs, which are not translated, either.
There are also numerous segments of DNA transcribed to RNA, which folds into complex and rather ornate structures, we know little about except they probably serve some vital function in the cell, and almost certainly errors in these may play role in disease. Here is a schematic of ITS-1 RNA coded for by rDNA:

I refer you to the extremely knowledgeable Adriana Heguy’s response to a similar question: What is a gene in modern biology?

If memory serves, the human genome contains ~17,500 polypeptide-coding genes, and about ~6,000 (known) genes that function is some other essential manner within the cell.

I expect we discover very few “new” protein-coding genes in humans because we already “know what to look for,” but will discover many more of the “other” varieties of genes over the next decade, or so. And yes, some of these are bound to turn up in intergenic regions, intronic sequences and maybe buried within the many thousands of pseudogenes our genomes are rife with.

Finally, to confuse the matter further, there are functional DNA/RNA hybrid species which have functional roles in cells. Four billion years of evolving life has generated all sorts of novel cellular machinery for us to sort out, but we’re getting there! Consider it took from 1990 to 2003 in giant labs to sequence the first human genome, at a cost of ~3 billion USD. Today it can be done on a bench in ONE DAY, and scientists are aiming for a cost of ~$1,000 USD, representing a cost reduction of over 6 orders of magnitude!

Images from: https://www.researchgate.net/figure/232718156_fig3_ITS1-rDNA-Folding-Structures-a-Predicted-RNA-folding-structure-of-ITS1-in-T-rotula