FASTQ files are generally raw reads, and you would not generally look for protein sequences from these raw reads. Possible reasons: short reads would only have fragments of the protein sequences on them, and long reads like pacbio/nanopore often have errors that make them not really suitable to use in their raw form (short reads also have errors). Either way, it's not common to look for protein sequences directly on reads to my knowledge.
Alternative approaches could include
1) (harder) use a de-novo genome assembler, to assemble your FASTQ reads into a genome assembly (FASTA), then, you would run gene prediction on your genome assembly. This would give you your protein sequences. doing a genome assembly is quite laborious. de-novo genome assemblers include miniasm, abyss, etc. gene prediction tools include MAKER
2) (easier) align the reads from the FASTQ file to a species that already has a reference genome e.g. http://hgdownload.soe.ucsc.edu/downloads.html#elephant (this is similar to what GenoMax suggests). and then you can check e.g. where is the P53 gene on the elephant genome, and how do the reads from the mammoth genome align in that region. you could determine the variants using a program called a variant caller. this process has caveats--you have to be careful interpreting the data, as you are aligning reads to a totally different species. but, generally in all data analysis, you have to be careful and be aware of caveats. aligning reads can be done with tools like bowtie, bwa, or minimap2. variant callers include tools like bcftools or gatk. you can then determine the effect of those variants using a variant effect prediction, this can tell you the consequence on the protein using tools like ensembl VEP or SnpEff.
It is a class project, so it isn't necessarily the optimal solution that is required.
it may help to have perspective on "proper" solutions on a bioinformatics forum anyways :)