Can immigration officers call another country to determine whether a traveller is a citizen of theirs?

In that case, if the immigration officers still had suspicion, could they call the country of the traveller to determine whether they are actually a citizen or not?

Probably not. Many countries do not keep a citizen's registry, at least not a complete one.

But the issuing authorities of machine-readable travel documents (e.g. passports) may be able to confirm if a travel document is issued by them (but in no way guaranteeing if the issuance was proper). Like a comment said, a genuine passport can still be fraudulently issued or used.

but I have heard of cases where the fake passports are manufactured by governments to the point where the security features appear to be indistinguishable from a real passport (e.g. Israel making fake Irish passport).

Electronic passports have a chip with an electronic signature that can be verified by public cryptography. If the chip is validly signed, it usually means it is a genuine passport.

If the private key is compromised... well, the database itself may also be compromised. In that case nothing but a complete investigation can resolve this.

The officers there believe the passport is fake but can't prove it beyond a reasonable doubt.

They do not have to prove it beyond a reasonable doubt, it's not a criminal court. Border officials are entitled to do a lot of things other government officials in other contexts may not be able to (search without warrants etc.), especially regarding foreigners.

While certain entitlements may exist for foreign citizens with particular status (e.g. EU citizens entering EU, permanent residents in certain countries), in most cases, the border official can refuse entry for any reason. Even if a reason is required or a non-arbitrary standard needs to be applied, the standard of proof is usually satisfied by an officer's good faith belief on reasonable grounds.

Do countries have that information?

The contact information of issuing authorities can be available via ICAO or Interpol, or by bilateral agreements.

ICAO Doc 9303 provides guidelines on the verification of machine-readable travel documents.

In particular, single points of contact should exist for countries participating in ICAO's LDS2 PKI program.

Additionally, it recommends

6.4.1 When a travel document gets a “hit” on INTERPOL’s lost, stolen or revoked database

A traveller should not be refused entry or prevented exit solely based on the document appearing on the lost, stolen or revoked travel document database. There are many steps that States must take to support these actions. If a traveller is in possession of a travel document that has been recorded as lost, stolen, or revoked on the ASF/SLTD, States should, where possible, liaise with the issuing and reporting country to confirm that the document has been rightfully recorded as a lost, stolen or revoked travel document. States should also conduct an interview with the traveller to ascertain his true identity or nationality, and determine if he is the rightful bearer of the travel document.

If the document contains a chip, States should conduct biometric verifications to support their efforts to determine the true identity of the traveller. States should also make efforts to determine whether the data have been altered and whether the document is authentic.

For the Interpol database, member states operate National Central Bureaus that may be able to deal with documentation questions. ICAO and Interpol also recommend member states to maintain a contact centre that can respond to other border authorities.

It is recommended that TDIAs make available to Interpol a 24/7 contact to confirm the status of reported documents and to resolve Hits in the Interpol database on a timely manner.

https://www.icao.int/Security/FAL/TRIP/Documents/Guide%20Part%201,%202%20and%203.%2027March2017.pdf

I think you are mixing up a few different concepts in your question, but I'll try my best to answer.

First of all, countries do cooperate and may also have routines in place to swiftly answer requests from other authorities. Since you particularly refer to 'Border Security Australia', I just recently saw an episode where Australian immigration officers quickly pulled out a New Zealanders criminal record, so those countries obviously have some kind of cooperation for information exchange.

But then you are talking about verifying the genuinity of a passport and wether or not a particular person is of a claimed citizenship and those are two quite different problems.

By now most countries issue biometric passports and with an embedded RFID chip, it is fairly easy to verify if the passport is genuine or not since the data on the chip has been electronically signed by a known issuer. Some authorities, like for example the Norwegian Police, do not only make the required information available to other countries' authorities, but to anyone. From the linked web page, anyone can download the required data to verify a Norwegian passport or id card and they are planning to offer a public service within the next year, for anyone to check if a Norwegian id document has been reported stolen.

That does of course not prevent certain countries from issuing obviously genuine passports to unentitled persons. A few years ago, there was for example a larger case in Bulgaria where corrupt state officials wrongfully granted Bulgarian citizenship to paying customers, eventually allowing these persons to regularly get Bulgarian passports. It is difficult to say if these passports should be considered genuine or not, but foreign authorities would for all practical purposes consider them genuine if they had been verified.