Python
Here's one way to do it in Python via list comprehension (see below for alternative shorter version).
$ python -c 'import sys;print([ l for i,l in enumerate(sys.stdin,1) if i==2][0].count("word"))' < input.txt                                          
3
$ cat input.txt
nothing here
word and another word, and one more word
last line
How this works:
- we run python interpreter with -cflag, were commands are contained within single quotes;
- the input file input.txtis redirected intostdinstream of the python interpreter via<shell operator. Hence we needsysmodule.
- Using list comprehension structure [something for item in something], we read lines of text fromsys.stdin.
- enumerate(sys.stdin,1)allows us to count enumerate the lines, i.e. with each iteration of list comprehension, we'll get the line of text into- lvariable and index into- ivariable starting the count at 1.
- The i==2will filter out only line which index equals to 2. That's how we know which line to extract.
- Thus as a result our list will contain only one item, and within the list its index is 0. So, we refer to that item as[<list comprehension stuff here>][0]. 
-The.count("word")is what actually does the job of counting. By definition it returns a number of non-overlapping  occurrences of a substring in a string.
- finally all of that stuff was contained within print()statement. So whatever number the.count()method returns will show up on screen.
Shorter version
The shorter way to do the same in Python would be to use readlines() method instead of list comprehension, and refer to specific item in the list that readlines() produces. Note, that readlines() produces a list, and lists in Python are 0-indexed, which means if you want to read line x, you should reference list item x-1. For instance,
$ python -c 'import sys;print(sys.stdin.readlines()[1].count("word"))' < input.txt       
3
sed+grep
Of course, we don't have to stick with scripting languages alone. sed and grep provide sufficient tools which we can use to suit our needs. With grep -c we can count occurrence of matched lines, so all we have to do is extract the specific line we need, and split all words in that line into separate lines. Like so:
$ sed -n  '2{s/ /\n/g;p}' input.txt | grep -c 'word'
3