You can answer "Klingons speak Klingon" or "Klingons speak the Klingon language", or either of those with "the Klingons" instead of just "Klingons".
But "Klingons speak Klingon language" doesn't work.
"language" is usually used a countable noun in English -- and essentially always when we speak of a particular language with an adjective such as "English" or "Klingon", rather than speaking about language as a general ability. So we say "a language" or "the language" but not just "language" when we are speaking about a particular one or a particular kind, e.g. "a Germanic language" or "the English language".
In clear and proper pronunciation all consonants should be pronounced all the time. So you make one tlh sound and then you make another tlh sound. There is no "sandhi" going on here. This is especially true of affricates like the tlh, even within a word. When speakers are being more lax, it's not unusual to hear gemination of fricatives and plosives, especially within a word.
Plural markers are not required in Klingon, so tlhIngan Hol could refer to the language of one Klingon or of multiple Klingons. tlhInganpu' Hol can only refer to the language of multiple Klingons.
Putting two nouns (or noun phrases) next to each other creates a genitive relationship. Perhaps the first noun owns or possesses the second noun (like HoD Duj, "the captain's ship"). Perhaps there is a relationship connecting one to the other (like SuvwI' targh, "the warrior's targ"). Perhaps the second noun is made of the first noun (like baS taj, "metal knife"). Or the source or origin of the first noun is the second noun (like tera' qoq, "a Terran robot"). Or maybe the second noun is contained within the first noun (like veng vaS'a', "the city's great hall"). We could probably even break some of those down into more specific categories, but I hope you get the general idea.
So tlhIngan Hol can mean the language of a specific Klingon as in, "the Klingon's language", or it can refer to the language of any non-specific Klingon as in, "a Klingon's language". Or since the plural markers are not required, it might refer to "the Klingons' language" or "Klingons' language". Or Hol could even be interpreted as plural and any of those could be "languages". It might also be using tlhIngan in an adjectival fashion as the source or original, in which case the best English translation is just "the Klingon language".
There are really many possible ways to write the translation that differ more in the minute details of saying or writing the term than in differences of meaning. All the different ways to write it in English mean almost the exact same thing. Is there really any difference between "the language of Klingons" and "the Klingon language"?
If the specific context of a discussion indicates that we are talking about a specific Klingon (or Klingons) or more than one language, then some of those variations might be good choices. However, in the generic context of just talking about some Klingon language, you should assume it's the one most common generic language called "Klingon" or "the Klingon language".