Dover ferry port will border force prevent four people travelling in one car

will we be prevented from traveling by car by border force?

Unlikely - if nothing else UK Border Force don't really give a rat's posterior, assuming they wouldn't be interesting in detaining you for any of the usual reasons (y'know guns, drugs, being terrorists - that sort of thing), whether you're allowed into France is the more pertinent question and as per the FCO you should be:

From 15 June 2020, travellers arriving in France from the UK and wider European Area (EU, Andorra, Holy See, Iceland, Liechtenstein, Monaco, Norway, San Marino and Switzerland) are no longer required to demonstrate their travel is essential or hold an international travel certificate.

So, you should be allowed in, and a car with 4 people in it is hardly noteworthy - but will you have to quarantine when you get there?

If you're travelling on or after the the 10th July then the travel corridor with France will be in effect, which means that you don't have to self-isolate upon your return from France. Why is this relevant?

Because the rules for entering France say:

However, for travellers arriving from European countries, whose authorities have decided, without coordination, to apply quarantine measures upon entry to their territory for travellers arriving from European countries, a voluntary quarantine will be requested in reciprocity [...] travellers arriving from the United Kingdom, regardless of their nationality, will be invited to respect a period of quarantine as soon as the British quarantine measures, announced this evening, enter into practical effect.

aka.. France were only requiring travelers from the UK to self-isolate while the UK asked those arriving from France to do so. Suggesting once the "travel corridor" rules go into effect this requirement will be lifted. But we'll probably be able to say with more certainty on the 10th July

Of course we can't say for sure that the rules that come into force in July will still be the same in August.

It is impossible to say what will happen in the future, even a few weeks in the future.

Quarantine in your own country does not mean you will be free from quarantine in another country.

Have a looksie at :

https://uk.ambafrance.org/COVID-19-UK-and-French-travel-and-quarantine-measures#s-t3-Travellers-entering-France-from-the-UK

"From Monday 25 May 2020, the French Government is putting in place a voluntary 14-day quarantine measure for travellers arriving from certain countries.

For travellers arriving from European countries whose authorities have decided, in an uncoordinated fashion, to apply 14-day quarantine measures to travellers entering their territory from European countries, a voluntary 14-day quarantine period will be reciprocally requested."