While I don't know what their supposed line of descent looked like, if they even bothered to construct one, this is extremely unlikely except in the loosest sense that everybody is a relative of one another and the Achaemenids are far enough back that, being nobility, they're probably ancestors of just about everyone in Iran today.

The reasons for this are as follows:

1. The Qajar tribe were Iranized Turkmen, not Persian or even ethnolinguistically Iranian.

2. The Achaemenids were mostly forgotten or deeply shrouded within oral traditions well over a thousand years before anybody had heard of the Qajars. The Sasanians traced their origins to esteemed "ahenag", forebears, which may have referred to a mythicized version of the Achaemenids, akin to the Kayanids of Persian oral tradition as recorded in the Shah Nameh and its most important source, the Khwaday-Namag.

3. By far the most popular king to trace your ancestry to in the Iran of Late Antiquity was actually Arsaces, the founder of the Parthian Empire. It is stated in a Western source that Arsaces traced his descent to Artaxerxes I or II, the two longest-reigning kings of the Achaemenid empire; even if this were the case this claim was almost certainly spurious, because Arsaces was from a tribal confederation of the Eurasian steppe, not from Iranian nobility. The last Arsacid claimants were the Karen family, who reigned in Khorasan until the mid-late 9th century. The Sasanians were actually notable for not making this claim.

So, essentially, no, just propaganda. That's not even addressing the fact that Alexander's conquest would have broken any line of continuous royalty...