Vi havde en kort diskussion om regningen, men bagefter blev vi hurtigt enige.

Vi havde is the simple past of at have and here it means we had.

• Vi havde en kort diskussion = We had a short discussion
• This is a very natural Danish way to say that a discussion took place.

Other possibilities are possible, but they mean slightly different things:

• Vi har haft en kort diskussion = We have had a short discussion
  This connects more directly to the present, like English have had.
• Vi diskuterede kort = We discussed briefly
  This focuses more on the action of discussing than on the discussion as a noun.

So Vi havde en kort diskussion is idiomatic and completely natural.

Because diskussion is a common gender noun in Danish, so its indefinite singular article is en.

• en diskussion = a discussion
• et hus = a house

Danish nouns are mainly divided into:

• en-words (common gender)
• et-words (neuter)

Since diskussion is an en-word, you say:

• en kort diskussion

not:

• et kort diskussion

Because kort is describing an indefinite singular en-word: en diskussion.

In Danish, adjectives usually change form depending on the noun phrase:

• en kort diskussion = indefinite singular, common gender
• et kort møde = indefinite singular, neuter
• korte diskussioner = plural
• den korte diskussion = definite

So in en kort diskussion, the basic adjective form kort is correct.

Regningen is the definite form of regning, meaning the bill.

Danish often adds the definite article to the end of the noun:

• en regning = a bill
• regningen = the bill

This is different from English, where the is a separate word.

So:

• om regningen = about the bill

If you wanted an indefinite meaning, it would be:

• om en regning = about a bill

Here om means about.

• en diskussion om regningen = a discussion about the bill

The preposition om has several meanings in Danish depending on context, such as:

• about
• around
• sometimes part of fixed expressions

But after nouns like diskussion, snak, or samtale, om often means about:

• en samtale om jobbet = a conversation about the job
• en diskussion om politik = a discussion about politics

Bagefter means afterwards, after that, or later on.

In this sentence:

• men bagefter blev vi hurtigt enige
• = but afterwards we quickly agreed

Difference:

• efter is usually a preposition: after
  • efter middagen = after dinner
• bagefter is more like an adverb: afterwards
  • Vi spiste, og bagefter gik vi hjem = We ate, and afterwards we went home

So here bagefter is the right choice because it refers back to the previous event.

Because Danish is a V2 language in main clauses. That means the finite verb normally comes in the second position.

In the basic order, you could say:

• Vi blev hurtigt enige bagefter

But when bagefter is moved to the front for emphasis or flow, the verb must still stay in second position:

• Bagefter blev vi hurtigt enige

So the order becomes:

1. bagefter
2. blev
3. vi

This is very common in Danish.

Because at blive enig means to become agreed or more naturally to come to an agreement.

So:

• vi blev enige = we agreed / we came to an agreement

This is a very common Danish expression. Danish often uses blive where English uses a more idiomatic verb.

Compare:

• Vi blev enige om prisen = We agreed on the price
• De blev hurtigt enige = They quickly reached agreement

If you said vi var enige, that would mean:

• we were in agreement
• This describes a state, not the process of reaching agreement.

So:

• vi var enige = the agreement already existed
• vi blev enige = they arrived at agreement

Because enige is the plural form of the adjective, and it refers to vi = we.

• jeg er enig = I agree
• du er enig = you agree
• vi er enige = we agree
• de er enige = they agree

In this sentence:

• vi blev hurtigt enige
• = we quickly agreed

Since vi is plural, enige is required.

Enig means in agreement or of the same opinion, but it is often completed with om when you say what the agreement is about.

Examples:

• Jeg er enig = I agree
• Jeg er enig med dig = I agree with you
• Vi er enige om planen = We agree on the plan

So:

• enig med nogen = agree with someone
• enig om noget = agree about/on something

In your sentence, the object of agreement is not explicitly stated after enige, but it is understood from the context: the discussion was about regningen, so they ended up agreeing about that issue.

Hurtigt is an adverb meaning quickly, and it modifies blev enige.

In:

• bagefter blev vi hurtigt enige

it comes after the subject vi and before enige. This is a very natural placement.

You may also see different placements depending on emphasis, but this one is standard and smooth.

Compare:

• Vi blev hurtigt enige = natural
• Hurtigt blev vi enige = possible, but more marked or literary
• Vi hurtigt blev enige = not correct in normal Danish

So the sentence uses a normal Danish adverb position.

Yes, you could say:

• Vi diskuterede regningen kort
• Vi diskuterede kort regningen is much less natural
• better: Vi diskuterede regningen i kort tid or Vi diskuterede kort regningen only in special emphasis, so usually avoid that

But there is a nuance:

• Vi havde en kort diskussion om regningen focuses on the discussion as an event
• Vi diskuterede regningen focuses on the action of discussing

Both are correct, but the original sentence sounds very natural and slightly more polished or narrative.

Men means but, and it shows contrast.

Here the contrast is:

• first, there was a discussion
• however, it did not remain a disagreement
• afterwards they quickly agreed

So men helps express the idea:

• there was some tension or difference of opinion,
• but it was resolved quickly.

Without men, the sentence would be more neutral and less contrastive.