Here's some sample data
library(ShortRead)
example(readFastq)
resulting in
> sread(rfq)
  A DNAStringSet instance of length 256
      width seq
  [1]    36 GGACTTTGTAGGATACCCTCGCTTTCCTTCTCCTGT
  [2]    36 GATTTCTTACCTATTAGTGGTTGAACAGCATCGGAC
  [3]    36 GCGGTGGTCTATAGTGTTATTAATATCAATTTGGGT
  [4]    36 GTTACCATGATGTTATTTCTTCATTTGGAGGTAAAA
  [5]    36 GTATGTTTCTCCTGCTTATCACCTTCTTGAAGGCTT
  ...   ... ...
[252]    36 GTTTAGATATGAGTCACATTTTGTTCATGGTAGAGT
[253]    36 GTTTTACAGACACCTAAAGCTACATCGTCAACGTTA
[254]    36 GATGAACTAAGTCAACCTCAGCACTAACCTTGCGAG
[255]    36 GTTTGGTTCGCTTTGAGTCTTCTTCGGTTCCGACTA
[256]    36 GCAATCTGCCGACCACTCGCGATTCAATCATGACTT
Supposing that my reverse primer sequence is 'TTTCC', that the primer sequence occurs uniquely, and that I'm interested in exact matches. Here I find matches
primer <- "TTTCC"
matches <- vmatchPattern(primer, sread(rfq), fixed=TRUE)
For simplicity I only look at those with a single match
idx <- lengths(matches) == 1L
I narrow the reads and sequences to the corresponding end() of the match
rfq[idx] <- narrow(rfq[idx], 1, unlist(end(matches)[idx]))
And verify that I've done a good job
> sread(rfq)[idx]
  A DNAStringSet instance of length 22
     width seq
 [1]    27 GGACTTTGTAGGATACCCTCGCTTTCC
 [2]    34 GTACGCTGGACTTTGTAGGATACCCTCGCTTTCC
 [3]    27 GGACTTTGTAGGATACCCTCGCTTTCC
 [4]    27 GCCACCATGATTATGACCAGTGTTTCC
 [5]    30 GGGAGGGTGTCAATCCTGACGGTTATTTCC
 ...   ... ...
[18]    30 GGACGCTCGACGCCATTAATAATGTTTTCC
[19]     7 GTTTTCC
[20]     7 GGTTTCC
[21]    19 GGAAAACACCAATCTTTCC
[22]    19 GGAAAGATTGGTGTTTTCC
> quality(rfq)[idx]
class: SFastqQuality
quality:
  A BStringSet instance of length 22
     width seq
 [1]    27 ]]]]]]]]]]]]Y]Y]]]]]]]]]]]]
 [2]    34 ]]]]]]]]]]]]]]]]Y]]P]RRTYYV][VZXHF
 [3]    27 ]]]]]]]]]]]]R]]]]]YYY]VT]RY
 [4]    27 ]]]]]]V]]]]]]]]YV]]]T]]]]]]
 [5]    30 ]]]]]]]]]]]VY]]]]]R]]]T]Y]Y]RV
 ...   ... ...
[18]    30 ]]]]]]]]]O]]]]O]]R]]V]]]Y]]]WZ
[19]     7 ]]]]]]]
[20]     7 ]]]]]]]
[21]    19 ]]]]]]]]]]]Y]]]]]]]
[22]    19 ]]]]]]]]]]]]]]]]]]]