The answers to your questions are, 'probably no', and 'definitely no'.
On the Greek side, the earliest known source to perhaps mention China is the Indika of Ktesias, which supposedly refers to the people of Seres as exceptionally tall and long-lived. This is, however, almost certainly not an original part of Ktesias. For one, our only source for the Indika of Ktesias is in a summary by the 9th century Byzantine patriarch Photios, rather than the original. For another, the passage referring to the Seres is an interpolation in one particular Latin manuscript of Photios, and so is extremely unlikely to be from Photios himself anyway. The Seres were a relatively popular subject for authors of the 1st century CE, particularly those writing in Latin, and it seems probable that a later medieval scribe chose to introduce apparent information about the Seres, derived from 1st century Latin sources, to their edition of Photios' summary of Ktesias.
Our earliest Greek source to definitely mention China is the Geographika of Strabo, whose first version was likely completed around 7 BCE (with a final update no earlier than 23 CE), over three centuries following Alexander's death. There are two mentions of the place in Strabo: first at 9.9, where he states that at its height, the Greco-Baktrian kingdom extended as far as Seres, and at 15.1, where in a section on India's botany, he mentions trees where wool grew on their branches, and notes that Serika was the same – it seems that he is describing silk garments, i.e. garments from Seres, rather than a variant form. While Strabo's main sources for India are the accounts of various of Alexander's followers, like Aristoboulos and Nearchos, whom he cites consistently, the reference to Serika at 15.1 is unsourced. The information about Baktria's borders, meanwhile, could have emerged only after the Greco-Baktrian kingdom had come into existence in the mid-3rd century BCE, around seventy years after Alexander.
The earliest Chinese reference to any sort of Hellenic polity is in the Histories (Shiji 史記) of Sima Qian 司馬遷, completed c. 94 BCE. Sima Qian's main source was an earlier traveller named Zhang Qian 張騫, who had been sent out west as an official envoy to broker an anti-Xiongnu alliance, but was captured and fled across the northern edge of the Tarim Basin, stayed for some time in the Ferghana Valley and Baktria, and returned to China via the southern route. His reports do not survive directly, but are drawn on and in part summarised both by Sima Qian and by the compilers of the Book of [Former] Han (han shu 漢書), completed c. 111 CE.
Two such polities are described in the summaries by Sima Qian: Daxia 大夏 in Baktria (modern-day Afghanistan) and Dayuan 大宛 in the Ferghana Valley (part of modern-day Uzbekistan and Tajikistan). The former was most probably referring to the Greco-Baktrian kingdom, while the latter seems to have been some kind of breakaway polity from the Greco-Baktrian kingdom, either a frontier region that rebelled or one that was cut off after the conquest of Baktria by the Yuezhi and Skythians. The etymology of Dayuan has been suggested to be the Persian Yauna ('Ionian'), generally used to denote Greeks, with da simply being 'great' and yuan being the ethnonym. By the time Zhang Qian was in Baktria, the Yuezhi had already conquered the state, but Dayuan was independent until its subjugation by the Han in the 'War of the Heavenly Horses' in 104-101 BCE. Zhang Qian also seems to have written a report on Tiaozhi 條支, a region commonly identified with the Seleukid Empire, but this appears only in the Book of Han's summary.
It must be noted that all of these references are somewhat indirect and even speculative. As far as Greek and Roman geographers were concerned, Seres was simply where silk came from, and the precise location given to Seres varies from the Red Sea to northern India to a region west of Baktria. The Chinese historians report only on regions of interest to the Han state – its vassal state of Dayuan in Ferghana, its neighbour Daxia, and the somewhat more distant state of Tiaozhi. They do not actually describe Greece itself, if we define Greece as mainland Greece, although we can definitely say they knew about the Greeks in general, to some extent – or more accurately, they knew about some Greeks.
But it will bear mentioning that in none of these cases were the two aware of each other during the lifetime of Alexander, and there is good reason for this – namely, in Alexander's time, the Warring States period in China was still ongoing, so there was no unified Chinese polity, let alone one with any presence in Central Asia. Alexander himself ventured only as far as the western tip of Ferghana, where he established an Alexandria (known both as Alexandria on the Tanais and Alexandria Eschate ('further Alexandria')). In other words, the furthest part of Alexander's empire in Central Asia was still some 3000km away from the westernmost significant settlement of a Chinese state. Alexander's interests lay northwest (towards the Skythian steppe) and southeast (towards India), not directly east across the Tarim Basin, and the Warring States had each other to deal with rather than try to find out about a region that only briefly appeared on the doorstep of their neighbours' neighbours.
But, if you want to discard all pretence of historical accuracy, then we can always have a look at an Arabic Alexander Romance. Qudāma ibn Ja'far, writing some time between 928 and 932, wrote that Alexander at one stage obtained the submission of both the Tibetans and the Chinese:
When they arrived on the border of China, the king of Tibet in the vanguard and Alexander following with the bulk of his army, the king of China came out to meet them at the head of ten battalions, each one numbering 100,000 men. He sent a message to Alexander, telling him that he had been informed of his good faith and noble actions, and because of this had no desire to fight, although he felt quite prepared to do so if Alexander wanted war. He asked Alexander to let him know which he preferred.
Alexander replied that the king of China must pay one-tenth of the revenues of his empire, just as the rulers of other countries had agreed to do when they submitted. If he refused, Alexander would not quail before the armies of the king, but put his trust in God, who had the power to make a small army triumph over a large. Along with this message he sent a number of Indians and Persians, whom he asked to bear witness to his justness and good treatment.
The king of China immediately made his submission and asked Alexander if he could pay the tribute in white and coloured silk and other manufactured goods. Alexander agreed, and the king sent him 1,000,000 pieces of coloured silk and the same of white, 500,000 of damask silk [kīmkāwa], 10,000 saddles with stirrups, bridles, cinches and so forth, along with 1,000,000 mann of silver.
Alexander stayed in the country until he had finished building a city, called Stone Tower [Burj al-Hijāra]. He garrisoned it with 5,000 Persians under the command of one of his generals named Neoclides.
Then he turned to the north, still accompanied by the king of China, and conquered the land of Shūl. Then he turned his attention to the Turks of the steppes, who submitted. There he learned that there was a very numerous Turkic people to the north-west, who were harassing neighbouring countries with their invasions. Alexander consulted the king of China about them, and the king told him that the only booty worth taking from them were herd animals and iron. He said their country was a remote corner of the earth with the Green Sea, which no one can sail, to the north and very high and inaccessible mountains to the west and south. In fact, there was only one way out, a narrow defile like a corridor. If this corridor could be closed, they couldn’t get out, and the world would be spared their ravages. Alexander immediately recognized the truth of this, and closed the defile with the wall spoken of in the Qur’ān.
Needless to say, this account is entirely fiction as far as the historical Alexander is concerned, who did not venture beyond the Ferghana Valley but instead turned south towards India. The Tibetan and Chinese states described are also basically calques of the then-semi-contemporary Tibetan and Tang empires (the latter had in fact collapsed in 906), rather than the actual state of affairs in the late fourth century BCE. But it is representative of a number of folkloric tales of Alexander that circulated in the Muslim world. Aside from the Tower of Alexander, there was also 'Alexander's Wall', supposedly built to contain the people of Gog and Magog, and hence also known as the 'Barrier of Gog and Magog'. The first references to this barrier come from the 7th century, and probably inspired the last portion of Qudāma ibn Ja'far's account above. According to Ibn Khurradādbih, relaying information gathered by an interpreter named Sallām in 844, the location of the Barrier was three days' march eastward from Hami, which could place it around the spot of the 'Jade Gate' (Yumen 玉門). This marked the western end of the Han-era 'Great Wall', long in disuse but with much of its ruins still standing.
But, this is a complete digression at this stage. No, the Greeks and Chinese did not know about each other before the 1st century AD.
Strabo, Geographika
Andrew Nichols (trans.), Ctesias: On India and Fragments of his Minor Works (2011)
Paul Lunde, Caroline Stone (eds.), Ibn Fadlan and the Land of Darkness: Arab Travellers in the Far North (2012)
Rachel Mairs (ed.), The Greco-Bactrian and Indo-Greek World (2020)