Using a simple shell script. If you're running windows, install cygwin.

gunzip -c fastq.gz |\
paste - - - - |\
awk '(!($2 ~ "ATGC"))' |\
tr "\t" "\n" |\
gzip > out.fastq.gz

• gunzip uncompress the source if needed.
• paste: convert 4 consecutive lines into one line with four columns *awk filter out column n°2 containing ATGC
• tr: convert tabulation to 'return' = convert back to fastq
• compress the output if needed and redirect to the output file "out.fastq.gz"

Python, using Biopython:

import sys
from Bio import SeqIO
sequences_of_interest = ["ATG", "GGCACAC", "GTGACCC"] #To be adapted
handle = open(sys.argv[1], "rU")
filtered = []
for record in SeqIO.parse(handle, "fastq")
    for seq in sequences_of_interest:        
        if record.seq.count(seq) > 0:
             filtered.append(record)
             break
output_handle = open("seqlected.fasta", "w")
SeqIO.write(filtered, output_handle, "fastq")
output_handle.close()
handle.close()

Save script (e.g. selectSEQ.py) and execute as python selectSEQ.py yourfile.fastq

Let me know if you need additional help.

There is FASTQ-GREP for filtering for sequences matching a pattern: http://homes.cs.washington.edu/~dcjones/fastq-tools/fastq-grep.html

If you are able to use Julia, you can use OpenGene(https://github.com/OpenGene/OpenGene.jl) to do this very easily

using OpenGene

instream = fastq_open("your.fq.gz")
outstream = fastq_open("out.fq.gz", "w")

while (fq = fastq_read(instream)) != false
    if contains(fq.sequence.seq, "Your pattern")
        fastq_write(outstream, fq)
    end
end

close(outstream)

If you have no idea about Julia, just do
1, run sudo apt-get install julia in ubuntu
2, run julia after the installation is finished
3, change the filename and your pattern in the code
4, paste your code into the Julia interactive command line and press ENTER